Anisotropic Spaces of Eight-Dimensional Numbers with Real Components

1. Introduction

Hypercomplex numbers generalize real and complex numbers to higher dimensions. As stated by the well-known Frobenius theorem [1], such generalizations require sacrificing some of the desirable properties of one-dimensional (real) and two-dimensional (complex) numbers. In practice, the choice has generally been between sacrificing commutativity of multiplication and admitting zero divisors. Zero divisors were commonly regarded as the greater disadvantage, and since Hamilton, researchers have therefore chosen to sacrifice commutativity. This led to quaternions, octonions, Clifford, Grassmann, and Pauli numbers [2,3], among others. Quaternions have found extensive applications in physics and mechanics [4,5] and are now among the principal tools used to solve certain classes of problems.

The Kazakh mathematician M. M. Abenov [6] followed a different route, regarding the presence of zero divisors as the lesser disadvantage. In 2019, he developed a four-dimensional mathematics distinct from the quaternionic framework [6], in which multiplication is associative and commutative but zero divisors are present. Abenov constructed a coherent theory, which he called four-dimensional mathematics, and, together with his colleagues, demonstrated several applications to hydrodynamic problems. Subsequently, Abenov and M. B. Gabbassov obtained other spaces of four-dimensional numbers in [7]. Particular spaces of four-dimensional numbers were considered in [8,9]. The resulting apparatus of four-dimensional mathematics proved sufficiently effective for solving various problems in fluid and gas mechanics [10,11,12]. The present work generalizes [13], where all anisotropic spaces of four-dimensional numbers were considered. We also mention [14], which studies four-dimensional spaces of numbers with complex components and applies them to the description of gates in two-qubit quantum systems.

This paper considers anisotropic spaces of eight-dimensional numbers with real components and investigates their properties. In these spaces, addition and multiplication are associative and commutative, while zero divisors are present. We show that the zero divisors form closed subspaces of the ambient space and satisfy specified relations among their components. A spectral norm of eight-dimensional numbers is defined, and its properties are established. The results show that eight-dimensional numbers naturally generalize four-dimensional, complex, and real numbers. The presence of zero divisors is a natural property of multidimensional number systems; in the complex case, the set of zero divisors collapses to the single zero element.

Some results of this study were presented at the 2025 international conference "Automorphisms and Identities of Algebras" [15].

2. Hypercomplex Numbers

Addition and multiplication of real numbers are commutative, associative, and distributive, and the real numbers have no zero divisors. The complex numbers constitute a natural extension of the real numbers. Addition and multiplication of complex numbers possess the same desirable properties as their real counterparts. Split-complex and dual numbers are also known; their addition is defined in the same way as for complex numbers, whereas multiplication is defined, respectively, by

$$\left(a+bi\right)\left(c+di\right)=ac+bd+\left(ad+bc\right)i,$$

(1)

$$\left(a+bi\right)\left(c+di\right)=ac+\left(ad+bc\right)i,$$

(2)

where $a+bi$ and $c+di$ are split-complex numbers in (1) and dual numbers in (2), respectively. The numbers a, b and c, d are called the components of the corresponding numbers, and i is the imaginary unit. In general, complex, split-complex, and dual numbers can be written as $a+bi$, where

$$i^2=\gamma .$$

(3)

In (3), $\gamma =-1$ for complex numbers, $\gamma =1$ for split-complex numbers, and $\gamma =0$ for dual numbers. Addition and multiplication are commutative and associative for all three types. The number $a-bi$ is called the conjugate of $a+bi$, and the real number $\sqrt{a^2+b^2}$ is called the norm of $a+bi$. The product $\left(a+bi\right)\left(a-bi\right)=a^2-\gamma b^2$ is real.

In the complex numbers, the equation $\left(a+bi\right)\left(x+yi\right)=c+di$ has the unique solution $x+yi={\displaystyle \frac{\left(a-bi\right)\left(c+di\right)}{a^2+b^2}}$. For split-complex and dual numbers, zero divisors arise, that is, nonzero numbers whose product is zero. The zero divisors among the split-complex numbers are the numbers of the form $a+ai$ and $a-ai$, $a\in R$, whereas among the dual numbers they are the numbers of the form $ai,a\in R$. Complex numbers have no zero divisors, which accounts for their extensive use in the natural sciences. The number systems just described are two-dimensional hypercomplex number systems.

Definition. An n-dimensional hypercomplex number is a multidimensional number of the form

$$x=a_0+a_1{i}_1+a_2{i}_2+\dots +a_{n-1}{i}_{n-1},$$

(4)

where $a_k\in R,k=0,1,\dots ,n-1$, $i_k,k=1,2,\dots ,n-1$ are imaginary units. An n-dimensional hypercomplex number can equivalently be represented by the vector $\left(a_0,a_1,a_2,\dots ,a_{n-1}\right)\in R^n$. Equality and addition are defined componentwise, while multiplication is determined by the multiplication rules for the imaginary units:

$$i_j{i}_k={\gamma }_{jk0}+{\gamma }_{jk1}{i}_1+{\gamma }_{jk2}{i}_2+\dots +{\gamma }_{jk,n-1}{i}_{n-1},$$

(5)

where ${\gamma }_{jkl}\in R,j,k=1,2,\dots ,n-1$.

For two-dimensional hypercomplex numbers, the rules in (5) reduce to (3). The next extension of two-dimensional hypercomplex numbers was provided by quaternions, introduced by the Irish mathematician W. Hamilton in 1843. Quaternions are four-dimensional hypercomplex numbers with three imaginary units i, j, and k, whose multiplication rules are

$$i^2=j^2=k^2=-1,ij=-ji=k,ik=-ki=-j,jk=-kj=i.$$

Quaternion multiplication is associative but noncommutative, and quaternions have no zero divisors. Quaternions have found extensive applications in mechanics and physics. However, the noncommutativity of multiplication has prevented the construction of a fully fledged analogue of classical mathematical analysis. Other possible multiplication laws for four-dimensional numbers remained unexplored for a long time, particularly because the well-known Frobenius theorem [1] states that the quaternions form the unique finite-dimensional real associative noncommutative division algebra.

Only at the beginning of the twenty-first century did the Kazakh mathematician M. Abenov publish a monograph [6] considering four-dimensional hypercomplex numbers whose imaginary units i, j, and k satisfy the multiplication rules

$$i^2=-j^2=k^2=-1,ij=ji=k,ik=ki=-j,jk=kj=i.$$

(6)

The resulting algebra is associative and commutative but contains zero divisors. Subsequently, M. Abenov and M. Gabbassov found all associative and commutative algebras over the real field considered within their construction and denoted them by $M_l,l=2,3,4,5,6,7$. The multiplication rules (6) for four-dimensional numbers correspond to the space $M_3$. All six four-dimensional algebras $M_l,l=2,3,4,5,6,7$ are described in [13]. As in the cases of split-complex and dual numbers, their zero divisors can be described explicitly; these are called degenerate numbers, and division by a nondegenerate number can be defined uniquely. Moreover, M. Abenov constructed a natural framework for mathematical analysis and topology and applied it to partial differential equations [10,11,12]. He called this new mathematical apparatus four-dimensional mathematics.

The best-known eight-dimensional hypercomplex numbers are the octonions, also called octaves, which can be written as

$$x=a_0+a_1{i}_1+a_2{i}_2+a_3{i}_3+a_4{i}_4+a_5{i}_5+a_6{i}_6+a_7{i}_7,$$

(7)

where $a_k\in R,k=0,1,\dots ,7$, and $i_k,k=1,2,\dots ,7$ are imaginary units. Their multiplication rules are given in Table 1.

Octonions are also called Cayley numbers or the Cayley algebra [2]. Octonion multiplication is neither commutative nor associative, but octonions have no zero divisors. According to the Frobenius theorem, the Cayley algebra is the unique finite-dimensional real nonassociative algebra without zero divisors.

Besides Cayley numbers, another example of eight-dimensional hypercomplex numbers is provided by Pauli numbers. They can also be represented in the form (7), and the multiplication rules for their imaginary units are given in Table 2.

Using Table 2, one can show that multiplication of Pauli numbers is associative but noncommutative and that these numbers have zero divisors.

We construct eight-dimensional hypercomplex numbers with associative and commutative multiplication. By the Frobenius theorem, these numbers necessarily have zero divisors. By analogy with Abenov’s four-dimensional numbers, the resulting number system will be called an eight-dimensional algebra.

3. Associative and Commutative Multiplication of Eight-Dimensional Numbers

We represent an eight-dimensional number (7) as $X=(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)$, where all components are real. Two numbers X and $Y=(y_1,y_2,y_3,y_4,y_5,y_6,y_7,y_8)$ are equal if $x_k=y_k$ for every k. Addition and subtraction are defined componentwise. We now introduce a multiplication operation that is both associative and commutative. In general, the product of two eight-dimensional numbers is written as follows (cf. the system of relations (5)):

$$\begin{array}{cc}\hfill z_1& ={\gamma }_{11}{x}_1{y}_1+{\gamma }_{22}{x}_2{y}_2+{\gamma }_{33}{x}_3{y}_3+{\gamma }_{44}{x}_4{y}_4\hfill \\ \hfill & +{\gamma }_{55}{x}_5{y}_5+{\gamma }_{66}{x}_6{y}_6+{\gamma }_{77}{x}_7{y}_7+{\gamma }_{88}{x}_8{y}_8,\hfill \\ \hfill z_2& ={\gamma }_{21}{x}_2{y}_1+{\gamma }_{12}{x}_1{y}_2+{\gamma }_{43}{x}_4{y}_3+{\gamma }_{34}{x}_3{y}_4\hfill \\ \hfill & +{\gamma }_{65}{x}_6{y}_5+{\gamma }_{56}{x}_5{y}_6+{\gamma }_{87}{x}_8{y}_7+{\gamma }_{78}{x}_7{y}_8,\hfill \\ \hfill z_3& ={\gamma }_{31}{x}_3{y}_1+{\gamma }_{42}{x}_4{y}_2+{\gamma }_{13}{x}_1{y}_3+{\gamma }_{24}{x}_2{y}_4\hfill \\ \hfill & +{\gamma }_{75}{x}_7{y}_5+{\gamma }_{86}{x}_8{y}_6+{\gamma }_{57}{x}_5{y}_7+{\gamma }_{68}{x}_6{y}_8,\hfill \\ \hfill z_4& ={\gamma }_{41}{x}_4{y}_1+{\gamma }_{32}{x}_3{y}_2+{\gamma }_{23}{x}_2{y}_3+{\gamma }_{14}{x}_1{y}_4\hfill \\ \hfill & +{\gamma }_{85}{x}_8{y}_5+{\gamma }_{76}{x}_7{y}_6+{\gamma }_{67}{x}_6{y}_7+{\gamma }_{58}{x}_5{y}_8,\hfill \\ \hfill z_5& ={\gamma }_{51}{x}_5{y}_1+{\gamma }_{62}{x}_6{y}_2+{\gamma }_{73}{x}_7{y}_3+{\gamma }_{84}{x}_8{y}_4\hfill \\ \hfill & +{\gamma }_{15}{x}_1{y}_5+{\gamma }_{26}{x}_2{y}_6+{\gamma }_{37}{x}_3{y}_7+{\gamma }_{48}{x}_4{y}_8,\hfill \\ \hfill z_6& ={\gamma }_{61}{x}_6{y}_1+{\gamma }_{52}{x}_5{y}_2+{\gamma }_{83}{x}_8{y}_3+{\gamma }_{74}{x}_7{y}_4\hfill \\ \hfill & +{\gamma }_{25}{x}_2{y}_5+{\gamma }_{16}{x}_1{y}_6+{\gamma }_{47}{x}_4{y}_7+{\gamma }_{38}{x}_3{y}_8,\hfill \\ \hfill z_7& ={\gamma }_{71}{x}_7{y}_1+{\gamma }_{82}{x}_8{y}_2+{\gamma }_{53}{x}_5{y}_3+{\gamma }_{64}{x}_6{y}_4\hfill \\ \hfill & +{\gamma }_{35}{x}_3{y}_5+{\gamma }_{46}{x}_4{y}_6+{\gamma }_{17}{x}_1{y}_7+{\gamma }_{28}{x}_2{y}_8,\hfill \\ \hfill z_8& ={\gamma }_{81}{x}_8{y}_1+{\gamma }_{72}{x}_7{y}_2+{\gamma }_{63}{x}_6{y}_3+{\gamma }_{54}{x}_5{y}_4\hfill \\ \hfill & +{\gamma }_{45}{x}_4{y}_5+{\gamma }_{36}{x}_3{y}_6+{\gamma }_{27}{x}_2{y}_7+{\gamma }_{18}{x}_1{y}_8.\hfill \end{array}$$

(8)

where $Z=\left(z_1,z_2,z_3,z_4,z_5,z_6,z_7,z_8\right)=X·Y$. Writing the eight equations $X·Y=Y·X$ and equating the corresponding coefficients of $x_j{y}_k,j,k=1,2,\dots ,8$, we obtain ${\gamma }_{jk}={\gamma }_{kj},j,k=1,2,\dots ,8$. Thus, of the 64 coefficients ${\gamma }_{jk}$, only 36 are distinct.

Next, let $T=(t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8)$ be an arbitrary eight-dimensional number and consider the associativity identity $\left(XY\right)T=X\left(YT\right)$. This identity must hold for all eight-dimensional numbers X, Y, and T. We therefore equate the corresponding coefficients of identical products $x_j{y}_k{t}_l$, where j, k, and l range from 1 to 8. This yields 294 distinct equations for the coefficients ${\gamma }_{jk}$. Equating the coefficients of $x_3{y}_3{t}_2$ in the equation for the second component gives ${\gamma }_{11}{\gamma }_{33}={\gamma }_{23}{\gamma }_{34}$, whence ${\gamma }_{33}={\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{11}}}$. Equating the coefficients of $x_4{y}_3{t}_2$ in the same component gives ${\gamma }_{11}{\gamma }_{44}={\gamma }_{24}{\gamma }_{34}$, whence ${\gamma }_{44}={\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{11}}}$. Equating the coefficients of $x_2{y}_2{t}_3$ in the equation for the third component gives ${\gamma }_{11}{\gamma }_{22}={\gamma }_{23}{\gamma }_{24}$, and hence ${\gamma }_{22}={\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{11}}}$. Equating the coefficients of $x_2{y}_5{t}_5$ in the equation for the second component yields ${\gamma }_{55}={\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{11}}}$. In the same way, we obtain ${\gamma }_{66}={\displaystyle \frac{{\gamma }_{26}{\gamma }_{56}}{{\gamma }_{11}}}$ (from the coefficients of $x_6{y}_6{t}_2$), ${\gamma }_{77}={\displaystyle \frac{{\gamma }_{27}{\gamma }_{78}}{{\gamma }_{11}}}$ (from the coefficients of $x_7{y}_7{t}_2$), and ${\gamma }_{88}={\displaystyle \frac{{\gamma }_{28}{\gamma }_{78}}{{\gamma }_{11}}}$ (from the coefficients of $x_2{y}_8{t}_8$). Analysis of the remaining equations shows that ${\gamma }_{11}={\gamma }_{12}={\gamma }_{13}={\gamma }_{14}={\gamma }_{15}={\gamma }_{16}={\gamma }_{17}={\gamma }_{18}$. Consequently, the multiplication law (8) takes the form

$$\begin{array}{cc}\hfill z_1& ={\gamma }_{11}{x}_1{y}_1+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{11}}}{x}_2{y}_2+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{11}}}{x}_3{y}_3+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{11}}}{x}_4{y}_4\hfill \\ & +{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{11}}}{x}_5{y}_5+{\displaystyle \frac{{\gamma }_{26}{\gamma }_{56}}{{\gamma }_{11}}}{x}_6{y}_6+{\displaystyle \frac{{\gamma }_{27}{\gamma }_{78}}{{\gamma }_{11}}}{x}_7{y}_7+{\displaystyle \frac{{\gamma }_{28}{\gamma }_{78}}{{\gamma }_{11}}}{x}_8{y}_8,\hfill \\ \hfill z_2& ={\gamma }_{11}{x}_2{y}_1+{\gamma }_{11}{x}_1{y}_2+{\gamma }_{34}{x}_4{y}_3+{\gamma }_{34}{x}_3{y}_4\hfill \\ & +{\gamma }_{56}{x}_6{y}_5+{\gamma }_{56}{x}_5{y}_6+{\gamma }_{78}{x}_8{y}_7+{\gamma }_{78}{x}_7{y}_8,\hfill \\ \hfill z_3& ={\gamma }_{11}{x}_3{y}_1+{\gamma }_{24}{x}_4{y}_2+{\gamma }_{11}{x}_1{y}_3+{\gamma }_{24}{x}_2{y}_4\hfill \\ & +{\gamma }_{57}{x}_7{y}_5+{\gamma }_{68}{x}_8{y}_6+{\gamma }_{57}{x}_5{y}_7+{\gamma }_{68}{x}_6{y}_8,\hfill \\ \hfill z_4& ={\gamma }_{11}{x}_4{y}_1+{\gamma }_{23}{x}_3{y}_2+{\gamma }_{23}{x}_2{y}_3+{\gamma }_{11}{x}_1{y}_4\hfill \\ & +{\gamma }_{58}{x}_8{y}_5+{\gamma }_{67}{x}_7{y}_6+{\gamma }_{67}{x}_6{y}_7+{\gamma }_{58}{x}_5{y}_8,\hfill \\ \hfill z_5& ={\gamma }_{11}{x}_5{y}_1+{\gamma }_{26}{x}_6{y}_2+{\gamma }_{37}{x}_7{y}_3+{\gamma }_{48}{x}_8{y}_4\hfill \\ & +{\gamma }_{11}{x}_1{y}_5+{\gamma }_{26}{x}_2{y}_6+{\gamma }_{37}{x}_3{y}_7+{\gamma }_{48}{x}_4{y}_8,\hfill \\ \hfill z_6& ={\gamma }_{11}{x}_6{y}_1+{\gamma }_{25}{x}_5{y}_2+{\gamma }_{38}{x}_8{y}_3+{\gamma }_{47}{x}_7{y}_4\hfill \\ & +{\gamma }_{25}{x}_2{y}_5+{\gamma }_{11}{x}_1{y}_6+{\gamma }_{47}{x}_4{y}_7+{\gamma }_{38}{x}_3{y}_8,\hfill \\ \hfill z_7& ={\gamma }_{11}{x}_7{y}_1+{\gamma }_{28}{x}_8{y}_2+{\gamma }_{35}{x}_5{y}_3+{\gamma }_{46}{x}_6{y}_4\hfill \\ & +{\gamma }_{35}{x}_3{y}_5+{\gamma }_{46}{x}_4{y}_6+{\gamma }_{11}{x}_1{y}_7+{\gamma }_{28}{x}_2{y}_8,\hfill \\ \hfill z_8& ={\gamma }_{11}{x}_8{y}_1+{\gamma }_{27}{x}_7{y}_2+{\gamma }_{36}{x}_6{y}_3+{\gamma }_{45}{x}_5{y}_4\hfill \\ & +{\gamma }_{45}{x}_4{y}_5+{\gamma }_{36}{x}_3{y}_6+{\gamma }_{27}{x}_2{y}_7+{\gamma }_{11}{x}_1{y}_8.\hfill \end{array}$$

(9)

We again write the eight equations $\left(XY\right)T=X\left(YT\right)$, now using the multiplication law (9), and equate the coefficients of identical products $x_j{y}_k{t}_l$. This gives 255 equations for the 28 unknown coefficients ${\gamma }_{jk}$. In the equation for the fifth component, equating the coefficients of $x_5{y}_2{t}_2$ gives ${\gamma }_{25}{\gamma }_{26}={\gamma }_{23}{\gamma }_{24}$, whence ${\gamma }_{26}={\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}$. In the equation for the eighth component, equating the coefficients of $x_5{y}_3{t}_2$ gives ${\gamma }_{27}{\gamma }_{35}={\gamma }_{23}{\gamma }_{45}$, whence ${\gamma }_{27}={\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}$. Similarly, in the equation for the seventh component, the coefficients of $x_5{y}_4{t}_2$ give ${\gamma }_{28}{\gamma }_{45}={\gamma }_{24}{\gamma }_{35}$, whence ${\gamma }_{28}={\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}$. Finally, in the equation for the sixth component, the coefficients of $x_5{y}_4{t}_3$ give ${\gamma }_{38}{\gamma }_{45}={\gamma }_{25}{\gamma }_{34}$, whence ${\gamma }_{38}={\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}$. Comparing the coefficients of $x_8{y}_3{t}_5$ in the equation for the second component yields ${\gamma }_{78}={\displaystyle \frac{{\gamma }_{38}{\gamma }_{56}}{{\gamma }_{35}}}={\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}$. Comparing the coefficients of $x_5{y}_5{t}_3$ and $x_4{y}_5{t}_6$ in the equation for the third component yields ${\gamma }_{57}={\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}$ and ${\gamma }_{68}={\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}$. Comparing the coefficients of $x_5{y}_5{t}_4$ and $x_6{y}_5{t}_3$ in the equation for the fourth component gives ${\gamma }_{58}={\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}$ and ${\gamma }_{67}={\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}$. Comparing the coefficients of $x_3{y}_3{t}_5$ and $x_4{y}_4{t}_5$ in the equation for the fifth component gives ${\gamma }_{37}={\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}$ and ${\gamma }_{48}={\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}$. Likewise, the equation for the seventh component yields ${\gamma }_{46}={\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}$, while the equation for the sixth component gives ${\gamma }_{47}={\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}$. Equating the coefficients of $x_3{y}_2{t}_5$ in the equation for the eighth component yields ${\gamma }_{36}={\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}$.

Substituting these values into (9), we obtain

$$\begin{array}{cc}\hfill z_1& ={\gamma }_{11}{x}_1{y}_1+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{11}}}{x}_2{y}_2+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{11}}}{x}_3{y}_3+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{11}}}{x}_4{y}_4\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{11}}}{x}_5{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{11}{\gamma }_{25}}}{x}_6{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{11}{\gamma }_{35}^2}}{x}_7{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{11}{\gamma }_{45}^2}}{x}_8{y}_8,\hfill \\ \hfill z_2& ={\gamma }_{11}{x}_2{y}_1+{\gamma }_{11}{x}_1{y}_2+{\gamma }_{34}{x}_4{y}_3+{\gamma }_{34}{x}_3{y}_4\hfill \\ \hfill & +{\gamma }_{56}{x}_6{y}_5+{\gamma }_{56}{x}_5{y}_6+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{x}_8{y}_7+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{x}_7{y}_8,\hfill \\ \hfill z_3& ={\gamma }_{11}{x}_3{y}_1+{\gamma }_{24}{x}_4{y}_2+{\gamma }_{11}{x}_1{y}_3+{\gamma }_{24}{x}_2{y}_4\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{x}_7{y}_5+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{x}_8{y}_6+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{x}_5{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{x}_6{y}_8,\hfill \\ \hfill z_4& ={\gamma }_{11}{x}_4{y}_1+{\gamma }_{23}{x}_3{y}_2+{\gamma }_{23}{x}_2{y}_3+{\gamma }_{11}{x}_1{y}_4\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{x}_8{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{x}_7{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{x}_6{y}_7+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{x}_5{y}_8,\hfill \\ \hfill z_5& ={\gamma }_{11}{x}_5{y}_1+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{x}_6{y}_2+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{x}_7{y}_3+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{x}_8{y}_4\hfill \\ \hfill & +{\gamma }_{11}{x}_1{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{x}_2{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{x}_3{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{x}_4{y}_8,\hfill \\ \hfill z_6& ={\gamma }_{11}{x}_6{y}_1+{\gamma }_{25}{x}_5{y}_2+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{x}_8{y}_3+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{x}_7{y}_4\hfill \\ \hfill & +{\gamma }_{25}{x}_2{y}_5+{\gamma }_{11}{x}_1{y}_6+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{x}_4{y}_7+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{x}_3{y}_8,\hfill \\ \hfill z_7& ={\gamma }_{11}{x}_7{y}_1+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{x}_8{y}_2+{\gamma }_{35}{x}_5{y}_3+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{x}_6{y}_4\hfill \\ \hfill & +{\gamma }_{35}{x}_3{y}_5+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{x}_4{y}_6+{\gamma }_{11}{x}_1{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{x}_2{y}_8,\hfill \\ \hfill z_8& ={\gamma }_{11}{x}_8{y}_1+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{x}_7{y}_2+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{x}_6{y}_3+{\gamma }_{45}{x}_5{y}_4\hfill \\ \hfill & +{\gamma }_{45}{x}_4{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{x}_3{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{x}_2{y}_7+{\gamma }_{11}{x}_1{y}_8.\hfill \end{array}$$

(10)

Substitution into the associativity identity $\left(XY\right)T=X\left(YT\right)$ verifies that it holds for all X, Y, and T. We have therefore proved the following theorem.

Theorem 1. 

The product of eight-dimensional numbers defined by (10) is associative and commutative for arbitrary nonzero real constants ${\gamma }_{11},{\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}$.

The theorem implies that there are infinitely many distinct associative and commutative products of eight-dimensional numbers with real components. Accordingly, there are infinitely many spaces of eight-dimensional numbers with associative and commutative addition and multiplication. Numbers of the form $A^{re}=\left(a,0,0,0,0,0,0,0\right)$, where $a\in R$, will be called real numbers. Consider the product of an eight-dimensional number X and a real number $A^{re}$. By (10), $X·A^{re}=a·{\gamma }_{11}·X$. Thus, $X·J_1={\gamma }_{11}·X$, where $J_1=(1,0,0,0,0,0,0,0)$ is the unit element. To ensure compatibility between addition and multiplication, we require $X·J_1=X$, and therefore set ${\gamma }_{11}=1$. Multiplication of an eight-dimensional number by a real number then amounts to multiplying each component by that real number. Hence, the general associative and commutative multiplication law for eight-dimensional numbers takes the form (11).

$$\begin{array}{cc}\hfill z_1& =x_1{y}_1+{\gamma }_{23}{\gamma }_{24}{x}_2{y}_2+{\gamma }_{23}{\gamma }_{34}{x}_3{y}_3+{\gamma }_{24}{\gamma }_{34}{x}_4{y}_4\hfill \\ \hfill & +{\gamma }_{25}{\gamma }_{56}{x}_5{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8{y}_8,\hfill \\ \hfill z_2& =x_2{y}_1+x_1{y}_2+{\gamma }_{34}{x}_4{y}_3+{\gamma }_{34}{x}_3{y}_4\hfill \\ \hfill & +{\gamma }_{56}{x}_6{y}_5+{\gamma }_{56}{x}_5{y}_6+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{x}_8{y}_7+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{x}_7{y}_8,\hfill \\ \hfill z_3& =x_3{y}_1+{\gamma }_{24}{x}_4{y}_2+x_1{y}_3+{\gamma }_{24}{x}_2{y}_4\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{x}_7{y}_5+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{x}_8{y}_6+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{x}_5{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{x}_6{y}_8,\hfill \\ \hfill z_4& =x_4{y}_1+{\gamma }_{23}{x}_3{y}_2+{\gamma }_{23}{x}_2{y}_3+x_1{y}_4\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{x}_8{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{x}_7{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{x}_6{y}_7+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{x}_5{y}_8,\hfill \\ \hfill z_5& =x_5{y}_1+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{x}_6{y}_2+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{x}_7{y}_3+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{x}_8{y}_4\hfill \\ \hfill & +x_1{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{x}_2{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{x}_3{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{x}_4{y}_8,\hfill \\ \hfill z_6& =x_6{y}_1+{\gamma }_{25}{x}_5{y}_2+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{x}_8{y}_3+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{x}_7{y}_4\hfill \\ \hfill & +{\gamma }_{25}{x}_2{y}_5+x_1{y}_6+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{x}_4{y}_7+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{x}_3{y}_8,\hfill \\ \hfill z_7& =x_7{y}_1+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{x}_8{y}_2+{\gamma }_{35}{x}_5{y}_3+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{x}_6{y}_4\hfill \\ \hfill & +{\gamma }_{35}{x}_3{y}_5+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{x}_4{y}_6+x_1{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{x}_2{y}_8,\hfill \\ \hfill z_8& =x_8{y}_1+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{x}_7{y}_2+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{x}_6{y}_3+{\gamma }_{45}{x}_5{y}_4\hfill \\ \hfill & +{\gamma }_{45}{x}_4{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{x}_3{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{x}_2{y}_7+x_1{y}_8.\hfill \end{array}$$

(11)

Each set of seven numbers $\left({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}\right)$ determines a particular multiplication law and is called a defining set of numbers.

Let $\left({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}\right)$ be a fixed defining set, and denote by $M({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56})$ the space of eight-dimensional numbers equipped with the addition, subtraction, and multiplication operations introduced above. The following eight-dimensional numbers $J_k=(0,\dots ,0,1,0,\dots ,0)$, with 1 in the k-th position, will be called the basis numbers. In particular, $J_1$ is a real number, while $J_k,k=2,3,\dots ,8$ correspond to the imaginary units. Every eight-dimensional number can therefore be expanded in terms of the basis numbers as

$$X=\left(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8\right)={\displaystyle \sum _{k=1}^8}{x}_k{J}_k.$$

(12)

We construct the multiplication table for the imaginary units (Table 3). It is symmetric because multiplication is commutative.

Comparison with Table 1 and Table 2 shows that these tables cannot coincide for any values of the defining set. In the Cayley algebra, the conjugate of $X=\left(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8\right)$ is defined as $X^{*}=(x_1,-x_2,-x_3,-x_4,-x_5,-x_6,-x_7,-x_8)$, whereas in the Pauli algebra it is $X^{*}=\left(x_1,x_2,x_3,x_4,-x_5,-x_6,-x_7,-x_8\right)$. These definitions are motivated by the desire to obtain the Euclidean norm of an eight-dimensional number from the quadratic form $X·X^{*}$. We define conjugation differently: the norm will be obtained not as the square root of a quadratic form but as the eighth root of a homogeneous form of degree eight. An analogous approach was used for spaces of four-dimensional numbers [6]. To this end, together with an eight-dimensional number X, consider the following seven numbers:

$$\begin{array}{c}\begin{array}{c}X(5,6,7,8)=(x_1,x_2,x_3,x_4,-x_5,-x_6,-x_7,-x_8)\\ X(3,4,5,6)=(x_1,x_2,-x_3,-x_4,-x_5,-x_6,x_7,x_8)\\ X(2,4,5,7)=(x_1,-x_2,x_3,-x_4,-x_5,x_6,-x_7,x_8)\end{array}\\ \begin{array}{c}X(2,3,5,8)=(x_1,-x_2,-x_3,x_4,-x_5,x_6,x_7,-x_8)\\ X(3,4,7,8)=(x_1,x_2,-x_3,-x_4,x_5,x_6,-x_7,-x_8)\end{array}\\ \begin{array}{c}X(2,4,6,8)=(x_1,-x_2,x_3,-x_4,x_5,-x_6,x_7,-x_8)\\ X(2,3,6,7)=(x_1,-x_2,-x_3,x_4,x_5,-x_6,-x_7,x_8)\end{array}\end{array}.$$

(13)

Consider the product $X·X(5,6,7,8)·X(3,4,5,6)·X(2,4,5,7)·X(2,3,5,8)·X(3,4,7,8)·X(2,4,6,8)·X(2,3,6,7)$. It was computed symbolically in Python and has the form $(M\left(X\right),0,0,0,0,0,0,0)$: all components except the first vanish, while the first component $M\left(X\right)$ is a homogeneous polynomial of degree eight containing 835 terms. We write this result as

$$X·X^{*}=M\left(X\right)·J_1.$$

(14)

where

$$\begin{array}{cc}\hfill X^{*}& =X(5,6,7,8)X(3,4,5,6)X(2,4,5,7)X(2,3,5,8)\hfill \\ \hfill & \times X(3,4,7,8)X(2,4,6,8)X(2,3,6,7).\hfill \end{array}$$

(15)

The eight-dimensional number $X^{*}$ is called the conjugate of X. If $M\left(X\right)$ is a nonnegative definite form, it is a natural candidate for the modulus of X. Because of its size, we do not display the full expression for $M\left(X\right)$. Instead, by analogy with four-dimensional numbers, we factor $M\left(X\right)$ into quadratic forms.

The conjugate of an eight-dimensional number has a cumbersome expression: each component is a homogeneous polynomial of degree seven containing 429 terms. The conjugates of the basis numbers are much simpler to compute:

$$J_1^{*}=J_1,J_2^{*}={\gamma }_{23}^3{\gamma }_{24}^3{J}_2,J_3^{*}={\gamma }_{23}^3{\gamma }_{34}^3{J}_3,J_4^{*}={\gamma }_{24}^3{\gamma }_{34}^3{J}_4,J_5^{*}={\gamma }_{25}^3{\gamma }_{56}^3{J}_5,$$

$$J_6^{*}={\displaystyle \frac{{\gamma }_{23}^3{\gamma }_{24}^3{\gamma }_{56}^3}{{\gamma }_{25}^3}}{J}_6,J_7^{*}={\displaystyle \frac{{\gamma }_{23}^3{\gamma }_{25}^3{\gamma }_{34}^3{\gamma }_{56}^3}{{\gamma }_{35}^6}}{J}_7,J_8^{*}={\displaystyle \frac{{\gamma }_{23}^3{\gamma }_{25}^3{\gamma }_{34}^3{\gamma }_{56}^3}{{\gamma }_{45}^6}}{J}_8.$$

4. Matrices of Eight-Dimensional Numbers

Introduce the operator $A:X\to A\left(X\right)$, which associates the following matrix $A\left(X\right)$ with each eight-dimensional number:

$$\left(\begin{array}{cccccccc}{x}_1& {\gamma }_{23}{\gamma }_{24}{x}_2& {\gamma }_{23}{\gamma }_{34}{x}_3& {\gamma }_{24}{\gamma }_{34}{x}_4& {\gamma }_{25}{\gamma }_{56}{x}_5& \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}{x}_6& \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}{x}_7& \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}{x}_8\\ x_2& x_1& {\gamma }_{34}{x}_4& {\gamma }_{34}{x}_3& {\gamma }_{56}{x}_6& {\gamma }_{56}{x}_5& \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}{x}_8& \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}{x}_7\\ x_3& {\gamma }_{24}{x}_4& x_1& {\gamma }_{24}{x}_2& \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}{x}_7& \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}{x}_8& \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}{x}_5& \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}{x}_6\\ x_4& {\gamma }_{23}{x}_3& {\gamma }_{23}{x}_2& x_1& \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}{x}_8& \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}{x}_7& \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}{x}_6& \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}{x}_5\\ x_5& \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}{x}_6& \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}{x}_7& \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}{x}_8& x_1& \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}{x}_2& \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}{x}_3& \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}{x}_4\\ x_6& {\gamma }_{25}{x}_5& \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}{x}_8& \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}{x}_7& {\gamma }_{25}{x}_2& x_1& \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}{x}_4& \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}{x}_3\\ x_7& \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}{x}_8& {\gamma }_{35}{x}_5& \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}{x}_6& {\gamma }_{35}{x}_3& \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}{x}_4& x_1& \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}{x}_2\\ x_8& \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}{x}_7& \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}{x}_6& {\gamma }_{45}{x}_5& {\gamma }_{45}{x}_4& \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}{x}_3& \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}{x}_2& x_1\end{array}\right).$$

The product of two eight-dimensional numbers X and Y can then be written as $X·Y=A\left(X\right)·Y$, where multiplication on the left-hand side denotes multiplication of eight-dimensional numbers according to (11), whereas multiplication on the right-hand side denotes multiplication of a matrix by a vector. The importance of this matrix becomes apparent when its determinant is calculated.

Theorem 2. 

Let $\left({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}\right)\in R^7$ be a defining set satisfying ${\gamma }_{23}{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{35}{\gamma }_{45}{\gamma }_{56}\ne 0$. Then the mapping $A:X\to A\left(X\right)$ is injective and surjective, and hence bijective.

The proof is immediate.

For subsequent transformations, it is convenient to write the matrix $A\left(X\right)$ in the form

$$A\left(X\right)=\left(\begin{array}{cc}{A}_{00}\left(X\right)& A_{11}\left(X\right)\\ A_{10}\left(X\right)& A_{01}\left(X\right)\end{array}\right),$$

(16)

where

$$A_{00}\left(X\right)=\left(\begin{array}{cc}\begin{array}{cc}{x}_1& {\gamma }_{23}{\gamma }_{24}{x}_2\\ x_2& x_1\end{array}& \begin{array}{cc}{\gamma }_{23}{\gamma }_{34}{x}_3& {\gamma }_{24}{\gamma }_{34}{x}_4\\ {\gamma }_{34}{x}_4& {\gamma }_{34}{x}_3\end{array}\\ \begin{array}{cc}{x}_3& {\gamma }_{24}{x}_4\\ x_4& {\gamma }_{23}{x}_3\end{array}& \begin{array}{cc}{x}_1& {\gamma }_{24}{x}_2\\ {\gamma }_{23}{x}_2& x_1\end{array}\end{array}\right),$$

(17)

$$A_{10}\left(X\right)=\left(\begin{array}{cc}\begin{array}{cc}{x}_5& {\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{x}_6\\ x_6& {\gamma }_{25}{x}_5\end{array}& \begin{array}{cc}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{x}_7& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{x}_8\\ {\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{x}_8& {\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{x}_7\end{array}\\ \begin{array}{cc}{x}_7& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{x}_8\\ x_8& {\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{x}_7\end{array}& \begin{array}{cc}{\gamma }_{35}{x}_5& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{x}_6\\ {\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{x}_6& {\gamma }_{45}{x}_5\end{array}\end{array}\right),$$

(18)

$$A_{01}\left(X\right)=\left(\begin{array}{cc}\begin{array}{cc}{x}_1& {\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{x}_2\\ {\gamma }_{25}{x}_2& x_1\end{array}& \begin{array}{cc}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{x}_3& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{x}_4\\ {\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{x}_4& {\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{x}_3\end{array}\\ \begin{array}{cc}{\gamma }_{35}{x}_3& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{x}_4\\ {\gamma }_{45}{x}_4& {\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{x}_3\end{array}& \begin{array}{cc}{x}_1& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{x}_2\\ {\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{x}_2& x_1\end{array}\end{array}\right),$$

(19)

$$A_{11}\left(X\right)=\left(\begin{array}{cc}\begin{array}{cc}{\gamma }_{25}{\gamma }_{56}{x}_5& {\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6\\ {\gamma }_{56}{x}_6& {\gamma }_{56}{x}_5\end{array}& \begin{array}{cc}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8\\ {\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{x}_8& {\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{x}_7\end{array}\\ \begin{array}{cc}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{x}_7& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{x}_8\\ {\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{x}_8& {\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{x}_7\end{array}& \begin{array}{cc}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{x}_5& {\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{x}_6\\ {\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{x}_6& {\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{x}_5\end{array}\end{array}\right).$$

(20)

Introduce the following notation:

$$\begin{array}{cc}\hfill {\sigma }_1& =sgn\left({\gamma }_{23}\right),{\sigma }_2=sgn\left({\gamma }_{24}\right),{\sigma }_3=sgn\left({\gamma }_{25}\right),{\sigma }_4=sgn\left({\gamma }_{34}\right),\hfill \\ \hfill {\sigma }_5& =sgn\left({\gamma }_{35}\right),{\sigma }_6=sgn\left({\gamma }_{45}\right),{\sigma }_7=sgn\left({\gamma }_{56}\right).\hfill \end{array}$$

We also use the notation ${\sigma }_{jk}={\sigma }_j{\sigma }_k,{\sigma }_{jkl}={\sigma }_j{\sigma }_k{\sigma }_l,{\sigma }_{jklm}={\sigma }_j{\sigma }_k{\sigma }_l{\sigma }_m$, where the indices range from 1 to 7.

Make the following change of variables:

$$\begin{array}{c}{y}_1=x_1,y_2=\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}{x}_2,y_3=\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}{x}_3,y_4=\sqrt{\left|{\gamma }_{24}{\gamma }_{34}\right|}{x}_4,\\ y_5=\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}{x}_5,y_6={\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{25}\right|}}}{x}_6,y_7={\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{\left|{\gamma }_{35}\right|}}{x}_7,y_8={\displaystyle \frac{\sqrt{\left|{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{\left|{\gamma }_{45}\right|}}{x}_8\end{array}.$$

(21)

Expressing the matrix $A\left(X\right)$ in terms of the variables $y_k$, we obtain

$$A\left(Y\right)=\left(\begin{array}{cc}{A}_{00}\left(Y\right)& A_{11}\left(Y\right)\\ A_{10}\left(Y\right)& A_{01}\left(Y\right)\end{array}\right),$$

(22)

where

$$A_{00}\left(Y\right)=\left(\begin{array}{cccc}{y}_1& {\sigma }_{12}\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}{y}_2& {\sigma }_{14}\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}{y}_3& {\sigma }_{24}\sqrt{\left|{\gamma }_{24}{\gamma }_{34}\right|}{y}_4\\ {\displaystyle \frac{y_2}{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}}}& y_1& {\displaystyle {\sigma }_4\frac{\sqrt{\left|{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{24}\right|}}{y}_4}& {\displaystyle {\sigma }_4\frac{\sqrt{\left|{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{23}\right|}}{y}_3}\\ {\displaystyle \frac{y_3}{\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}}& {\displaystyle {\sigma }_2\frac{\sqrt{\left|{\gamma }_{24}\right|}}{\sqrt{\left|{\gamma }_{34}\right|}}{y}_4}& y_1& {\displaystyle {\sigma }_2\frac{\sqrt{\left|{\gamma }_{24}\right|}}{\sqrt{\left|{\gamma }_{23}\right|}}{y}_2}\\ {\displaystyle \frac{y_4}{\sqrt{\left|{\gamma }_{24}{\gamma }_{34}\right|}}}& {\displaystyle {\sigma }_1\frac{\sqrt{\left|{\gamma }_{23}\right|}}{\sqrt{\left|{\gamma }_{34}\right|}}{y}_3}& {\displaystyle {\sigma }_1\frac{\sqrt{\left|{\gamma }_{23}\right|}}{\sqrt{\left|{\gamma }_{24}\right|}}{y}_2}& y_1\end{array}\right),$$

(23)

$$A_{10}\left(Y\right)=\left(\begin{array}{cccc}{\displaystyle \frac{y_5}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}}& {\displaystyle \frac{{\sigma }_{123}\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{y}_6}& {\displaystyle \frac{{\sigma }_{14}\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{y}_7}& {\displaystyle \frac{{\sigma }_{24}\sqrt{\left|{\gamma }_{24}{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{y}_8}\\ {\displaystyle \frac{\sqrt{\left|{\gamma }_{25}\right|}{y}_6}{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}\right|}}}& {\displaystyle {\sigma }_3\sqrt{\frac{{\gamma }_{25}}{{\gamma }_{56}}}{y}_5}& {\displaystyle \frac{{\sigma }_{34}\sqrt{\left|{\gamma }_{25}{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{24}{\gamma }_{56}\right|}}{y}_8}& {\displaystyle \frac{{\sigma }_{34}\sqrt{\left|{\gamma }_{25}{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{23}{\gamma }_{56}\right|}}{y}_7}\\ {\displaystyle \frac{{\sigma }_5\left|{\gamma }_{35}\right|y_7}{\sqrt{\left|{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}}& {\displaystyle \frac{{\sigma }_{25}\sqrt{\left|{\gamma }_{24}\right|}\left|{\gamma }_{35}\right|}{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{y}_8}& {\displaystyle \frac{{\sigma }_5\left|{\gamma }_{35}\right|}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{y}_5}& {\displaystyle \frac{{\sigma }_{235}\sqrt{\left|{\gamma }_{24}\right|}\left|{\gamma }_{35}\right|}{\sqrt{\left|{\gamma }_{23}{\gamma }_{25}{\gamma }_{56}\right|}}{y}_6}\\ {\displaystyle \frac{{\sigma }_6\left|{\gamma }_{45}\right|y_8}{\sqrt{\left|{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}}& {\displaystyle \frac{{\sigma }_{16}\sqrt{\left|{\gamma }_{23}\right|}\left|{\gamma }_{45}\right|}{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{y}_7}& {\displaystyle \frac{{\sigma }_{136}\sqrt{\left|{\gamma }_{23}\right|}\left|{\gamma }_{45}\right|}{\sqrt{\left|{\gamma }_{24}{\gamma }_{25}{\gamma }_{56}\right|}}{y}_6}& {\displaystyle \frac{{\sigma }_6\left|{\gamma }_{45}\right|}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{x}_5}\end{array}\right),$$

(24)

$$A_{01}\left(Y\right)=\left(\begin{array}{cc}\begin{array}{cc}{y}_1& {\displaystyle \frac{{\sigma }_{123}\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}}{\left|{\gamma }_{25}\right|}}{y}_2\\ {\displaystyle \frac{{\sigma }_3\left|{\gamma }_{25}\right|}{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}}}{y}_2& y_1\end{array}& \begin{array}{cc}{\displaystyle \frac{{\sigma }_{145}\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}{\left|{\gamma }_{35}\right|}}{y}_3& {\displaystyle \frac{{\sigma }_{246}\sqrt{\left|{\gamma }_{24}{\gamma }_{34}\right|}}{\left|{\gamma }_{45}\right|}}{y}_4\\ {\sigma }_{345}{\displaystyle \frac{\left|{\gamma }_{25}\right|\sqrt{\left|{\gamma }_{34}\right|}}{\left|{\gamma }_{35}\right|\sqrt{\left|{\gamma }_{24}\right|}}}{y}_4& {\sigma }_{346}{\displaystyle \frac{\left|{\gamma }_{25}\right|\sqrt{\left|{\gamma }_{34}\right|}}{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}}{y}_3\end{array}\\ \begin{array}{cc}{\displaystyle \frac{{\sigma }_5\left|{\gamma }_{35}\right|}{\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}}{y}_3& {\sigma }_{235}{\displaystyle \frac{\left|{\gamma }_{35}\right|\sqrt{\left|{\gamma }_{24}\right|}}{\left|{\gamma }_{25}\right|\sqrt{\left|{\gamma }_{34}\right|}}}{y}_4\\ {\displaystyle \frac{{\sigma }_6\left|{\gamma }_{45}\right|}{\sqrt{\left|{\gamma }_{24}{\gamma }_{34}\right|}}}{y}_4& {\sigma }_{136}{\displaystyle \frac{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}{\left|{\gamma }_{25}\right|\sqrt{\left|{\gamma }_{34}\right|}}}{y}_3\end{array}& \begin{array}{cc}{y}_1& {\sigma }_{256}{\displaystyle \frac{\left|{\gamma }_{35}\right|\sqrt{\left|{\gamma }_{24}\right|}}{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}}{y}_2\\ {\sigma }_{156}{\displaystyle \frac{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}{\left|{\gamma }_{35}\right|\sqrt{\left|{\gamma }_{24}\right|}}}{y}_2& y_1\end{array}\end{array}\right),$$

(25)

$$A_{11}\left(Y\right)=\left(\begin{array}{cccc}{\sigma }_{37}\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}{y}_5& {\sigma }_{1237}{\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{25}\right|}}}{y}_6& {\sigma }_{1347}{\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{\left|{\gamma }_{35}\right|}}{y}_7& {\sigma }_{2347}{\displaystyle \frac{\sqrt{\left|{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{\left|{\gamma }_{45}\right|}}{y}_8\\ {\sigma }_7{\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}}}{y}_6& {\sigma }_7{\displaystyle \frac{\sqrt{\left|{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{25}\right|}}}{y}_5& {\sigma }_{34567}{\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{\left|{\gamma }_{35}\right|\sqrt{\left|{\gamma }_{24}\right|}}}{y}_8& {\sigma }_{34567}{\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}}{y}_7\\ {\sigma }_{37}{\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}}{y}_7& {\sigma }_{27}{\displaystyle \frac{\sqrt{\left|{\gamma }_{24}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}\right|}}}{y}_8& {\sigma }_{357}{\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\left|{\gamma }_{35}\right|}}{y}_5& {\sigma }_{267}{\displaystyle \frac{\sqrt{\left|{\gamma }_{24}{\gamma }_{25}{\gamma }_{56}\right|}}{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}}{y}_6\\ {\sigma }_{37}{\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{24}{\gamma }_{34}\right|}}}{y}_8& {\sigma }_{17}{\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}\right|}}}{y}_7& {\sigma }_{157}{\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{25}{\gamma }_{56}\right|}}{\left|{\gamma }_{35}\right|\sqrt{\left|{\gamma }_{24}\right|}}}{y}_6& {\sigma }_{367}{\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\left|{\gamma }_{45}\right|}}{y}_5\end{array}\right).$$

(26)

Choosing the signs ${\sigma }_i,i=1,2,3,4,5,6,7$ amounts to considering different sign combinations of the defining numbers. There are $2^7=128$ such combinations, and each gives a different expression for $M\left(X\right)$ in (14). These combinations can be divided into the following groups:

1.

All generating numbers are positive (1 case);

2.

One of the numbers ${\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6,{\alpha }_7,{\alpha }_8$ is negative and the remaining generating numbers are positive (7 cases);

3.

Two of the numbers ${\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6,{\alpha }_7,{\alpha }_8$ are negative and the remaining generating numbers are positive (21 cases);

4.

Three of the numbers ${\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6,{\alpha }_7,{\alpha }_8$ are negative and the remaining generating numbers are positive (35 cases);

5.

Four of the numbers ${\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6,{\alpha }_7,{\alpha }_8$ are negative and the remaining generating numbers are positive (35 cases);

6.

Five of the numbers ${\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6,{\alpha }_7,{\alpha }_8$ are negative and the remaining generating numbers are positive (21 cases);

7.

Six of the numbers ${\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6,{\alpha }_7,{\alpha }_8$ are negative and the remaining generating number is positive (7 cases);

8.

All generating numbers are negative (1 case).

Consider the first case, in which all generating numbers are positive, that is, ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4={\sigma }_5={\sigma }_6={\sigma }_7=1$. We calculate the determinant of $A\left(X\right)$. Apply the following transformations to $A\left(Y\right)$: subtract the second row multiplied by $\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}$ from the first row; subtract the fourth row multiplied by ${\displaystyle \frac{\sqrt{\left|{\gamma }_{24}\right|}}{\sqrt{\left|{\gamma }_{23}\right|}}}$ from the third row; subtract the sixth row multiplied by ${\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}}{\left|{\gamma }_{25}\right|}}$ from the fifth row; and subtract the eighth row multiplied by ${\sigma }_{56}{\displaystyle \frac{\left|{\gamma }_{35}\right|\sqrt{\left|{\gamma }_{24}\right|}}{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}}$ from the seventh row. Next, in the resulting matrix, add the first column multiplied by $\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}$ to the second column; add the third column multiplied by ${\displaystyle \frac{\sqrt{\left|{\gamma }_{24}\right|}}{\sqrt{\left|{\gamma }_{23}\right|}}}$ to the fourth column; add the fifth column multiplied by ${\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}}{\left|{\gamma }_{25}\right|}}$ to the sixth column; and add the seventh column multiplied by ${\sigma }_{56}{\displaystyle \frac{\left|{\gamma }_{35}\right|\sqrt{\left|{\gamma }_{24}\right|}}{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}}$ to the eighth column. Then interchange the second and fifth rows and the fourth and seventh rows, as well as the second and fifth columns and the fourth and seventh columns. After these transformations, $A\left(Y\right)$ becomes a block matrix in which $A_{11}\left(Y\right)$ is the zero matrix and

$$A_{00}\left(Y\right)=\left(\begin{array}{cccc}{y}_1-y_2& \sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}(y_5-y_6)& \sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}(y_3-y_4)& {\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{\left|{\gamma }_{35}\right|}(y_7-y_8)}\\ {\displaystyle \frac{y_5-y_6}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}}& y_1-y_2& {\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}(y_7-y_8)}& {\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}{\left|{\gamma }_{35}\right|}(y_3-y_4)}\\ {\displaystyle \frac{y_3-y_4}{\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}}& {\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}(y_7-y_8)}& y_1-y_2& {\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\left|{\gamma }_{35}\right|}(y_5-y_6)}\\ {\displaystyle \frac{\left|{\gamma }_{35}\right|(y_7-y_8)}{\sqrt{\left|{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}}& {\displaystyle \frac{\left|{\gamma }_{35}\right|}{\sqrt{\left|{\gamma }_{23}{\gamma }_{34}\right|}}(y_3-y_4)}& {\displaystyle \frac{\left|{\gamma }_{35}\right|}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}(y_5-y_6)}& y_1-y_2\end{array}\right).$$

$$A_{01}\left(Y\right)=\left(\begin{array}{cccc}{\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{23}{\gamma }_{24}\right|}}{y}_6}& y_1+y_2& {\displaystyle \frac{\sqrt{\left|{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{23}\right|}}(y_3+y_4)}& {\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}(y_7+y_8)}\\ {\displaystyle \sqrt{\frac{{\gamma }_{25}}{{\gamma }_{56}}}(y_5+y_6)}& y_1+y_2& {\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}\right|}}{\sqrt{\left|{\gamma }_{23}{\gamma }_{56}\right|}}(y_7+y_8)}& {\displaystyle \frac{\left|{\gamma }_{25}\right|\sqrt{\left|{\gamma }_{34}\right|}}{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}(y_3+y_4)}\\ {\displaystyle \frac{\sqrt{\left|{\gamma }_{23}\right|}}{\sqrt{\left|{\gamma }_{34}\right|}}(y_3+y_4)}& {\displaystyle \frac{\sqrt{\left|{\gamma }_{23}{\gamma }_{56}\right|}}{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}\right|}}(y_7+y_8)}& y_1+y_2& {\displaystyle \frac{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}{\left|{\gamma }_{45}\right|}(y_5+y_6)}\\ {\displaystyle \frac{\sqrt{\left|{\gamma }_{23}\right|}\left|{\gamma }_{45}\right|}{\sqrt{\left|{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}\right|}}(y_7+y_8)}& {\displaystyle \frac{\left|{\gamma }_{45}\right|\sqrt{\left|{\gamma }_{23}\right|}}{\left|{\gamma }_{25}\right|\sqrt{\left|{\gamma }_{34}\right|}}(y_3+y_4)}& {\displaystyle \frac{\left|{\gamma }_{45}\right|}{\sqrt{\left|{\gamma }_{25}{\gamma }_{56}\right|}}(x_5+y_6)}& y_1+y_2\end{array}\right).$$

The determinant of $A\left(Y\right)$ is the product of the determinants of $A_{00}\left(Y\right)$ and $A_{01}\left(Y\right)$. Computing the latter determinants gives

$$\begin{array}{cc}\hfill detA\left(Y\right)& =\left[{(y_1-y_2-y_3+y_4)}^2-{(y_5-y_6-y_7+y_8)}^2\right]\hfill \\ \hfill & \times \left[{(y_1-y_2+y_3-y_4)}^2-{(y_5-y_6+y_7-y_8)}^2\right]\hfill \\ \hfill & \times \left[{(y_1+y_2-y_3-y_4)}^2-{(y_5+y_6-y_7-y_8)}^2\right]\hfill \\ \hfill & \times \left[{(y_1+y_2+y_3+y_4)}^2-{(y_5+y_6+y_7+y_8)}^2\right].\hfill \end{array}$$

Thus, this determinant is not a positive definite form.

Consider the first case of the second group, in which ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4={\sigma }_5={\sigma }_6=1,{\sigma }_7=-1$. Here, the matrices $A_{00}\left(Y\right),A_{10}\left(Y\right)$ and $A_{01}\left(Y\right)$ are the same as in the first case, while $A_{11}\left(Y\right)$ changes sign. Applying transformations analogous to those used in the first case, we obtain

$$\begin{array}{cc}\hfill detA\left(Y\right)& =\left[{(y_1-y_2-y_3+y_4)}^2+{(y_5-y_6-y_7+y_8)}^2\right]\hfill \\ \hfill & \times \left[{(y_1-y_2+y_3-y_4)}^2+{(y_5-y_6+y_7-y_8)}^2\right]\hfill \\ \hfill & \times \left[{(y_1+y_2-y_3-y_4)}^2+{(y_5+y_6-y_7-y_8)}^2\right]\hfill \\ \hfill & \times \left[{(y_1+y_2+y_3+y_4)}^2+{(y_5+y_6+y_7+y_8)}^2\right].\hfill \end{array}$$

(27)

This is a nonnegative definite form of degree eight in the variables $y_i,i=1,2,\dots ,8$. Returning to the variables $x_i,i=1,2,\dots ,8$, we obtain

$$\begin{array}{c}detA\left(X\right)=[{\left(x_1-\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3+\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)}^2\hfill \\ +{\left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5-\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7+{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)}^2]\\ \times [{\left(x_1-\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)}^2\\ +{\left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5-\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)}^2]\\ \times [{\left(x_1+\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)}^2\\ +{\left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5+\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)}^2]\\ \times [{\left(x_1+\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3+\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)}^2\\ \hfill +{\left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5+\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7+{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)}^2].\end{array}$$

(28)

The determinant of $A\left(X\right)$ was computed for all 128 cases listed above. It is a nonnegative definite form in 112 cases and is not sign-definite in the remaining 16 cases. The sign patterns ${\sigma }_i,i=1,2,\dots ,7$ for which the determinant is not sign-definite are listed in Table 4.

Each ${\sigma }_i,i=1,2,\dots ,7$ equals +1 in eight cases and -1 in eight cases. The remaining 112 matrices, whose determinants are nonnegative definite forms of degree eight, are listed in Appendix A.

5. Spaces of Eight-Dimensional Numbers with Real Components

We now examine the relationship between spaces of eight-dimensional numbers and the matrix spaces introduced above. As shown earlier, the product of an eight-dimensional number (13) and X is a real number whose first component equals $M\left(X\right)$. Direct calculation shows that the determinant (28) of $A\left(X\right)$ coincides with $M\left(X\right)$ in (14). This equality was verified symbolically in Python for every case. Thus, among the 128 possible sign combinations of the defining numbers, $M\left(X\right)$ is a nonnegative definite form in 112 cases. In these cases, $\sqrt[8]{M\left(X\right)}$ will be called the symplectic modulus of X. All combinations ${\sigma }_i,i=1,2,\dots ,7$ for which the modulus of an eight-dimensional number is defined are listed in Appendix A. Each corresponding space of eight-dimensional numbers will be denoted by $M\left({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}\right)$, or by $M\left({\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4,{\sigma }_5,{\sigma }_6,{\sigma }_7\right)$ when the particular values of the defining numbers are immaterial, and more briefly by $M_k,k=1,2,\dots ,112$, where k is the case number in Appendix A. The spaces defined in this way will be called anisotropic spaces of eight-dimensional numbers with real components.

For example, expression (28) defines the symplectic modulus in the space $M({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56})$ when ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4={\sigma }_5={\sigma }_6=1,{\sigma }_7=-1$. Naturally, all defining numbers are assumed to be strictly positive within each space.

A space in which all defining numbers are equal to one is called an isotropic space of eight-dimensional numbers with real components.

An important feature distinguishing these spaces from analogous systems such as the octonions is that both addition and multiplication are commutative and associative. On the other hand, these spaces contain zero divisors, which are absent from spaces equipped with a Euclidean norm. Zero divisors are considered in greater detail in the next section.

Theorem 3. 

For every space $M\left({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}\right)$, the mapping $A:x\to A\left(x\right)$ has the following properties for arbitrary eight-dimensional numbers $x,y$:

1. 

$A\left(x\pm y\right)=A\left(x\right)\pm A\left(y\right)$;

2. 

$A\left(cx\right)=cA\left(x\right)$ for every $c\in R$;

3. 

$A\left(xy\right)=A\left(x\right)A\left(y\right)$;

4. 

$det\left(A\left(x\right)\right)={\left|x\right|}^8$;

5. 

$det\left(A\left(x\right)\pm A\left(y\right)\right)={\left|x\pm y\right|}^8$;

6. 

$det\left(A\left(\alpha x\right)\right)={\left|\alpha x\right|}^8$;

7. 

$det\left(A\left(x\right)F\left(y\right)\right)={\left|xy\right|}^8$;

where $\left|x\right|$ is the symplectic modulus of the eight-dimensional number x.

Proof. 

It suffices to prove the assertion for $M_1=M(1,1,1,1,1,1,-1)$; the other spaces are treated analogously.

Properties 1 and 2 are immediate. We prove Property 3. The product $XY=Z$ in $M_1$ is defined by

$$\begin{array}{cc}\hfill z_1& =x_1{y}_1+{\gamma }_{23}{\gamma }_{24}{x}_2{y}_2+{\gamma }_{23}{\gamma }_{34}{x}_3{y}_3+{\gamma }_{24}{\gamma }_{34}{x}_4{y}_4\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5{y}_5-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6{y}_6-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7{y}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8{y}_8,\hfill \\ \hfill z_2& =x_2{y}_1+x_1{y}_2+{\gamma }_{34}{x}_4{y}_3+{\gamma }_{34}{x}_3{y}_4\hfill \\ \hfill & -{\gamma }_{56}{x}_6{y}_5-{\gamma }_{56}{x}_5{y}_6-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{x}_8{y}_7-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{x}_7{y}_8,\hfill \\ \hfill z_3& =x_3{y}_1+{\gamma }_{24}{x}_4{y}_2+x_1{y}_3+{\gamma }_{24}{x}_2{y}_4\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{x}_7{y}_5-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{x}_8{y}_6-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{x}_5{y}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{x}_6{y}_8,\hfill \\ \hfill z_4& =x_4{y}_1+{\gamma }_{23}{x}_3{y}_2+{\gamma }_{23}{x}_2{y}_3+x_1{y}_4\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{x}_8{y}_5-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{x}_7{y}_6-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{x}_6{y}_7-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{x}_5{y}_8,\hfill \\ \hfill z_5& =x_5{y}_1+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{x}_6{y}_2+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{x}_7{y}_3+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{x}_8{y}_4\hfill \\ \hfill & +x_1{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{x}_2{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{x}_3{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{x}_4{y}_8,\hfill \\ \hfill z_6& =x_6{y}_1+{\gamma }_{25}{x}_5{y}_2+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{x}_8{y}_3+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{x}_7{y}_4\hfill \\ \hfill & +{\gamma }_{25}{x}_2{y}_5+x_1{y}_6+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{x}_4{y}_7+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{x}_3{y}_8,\hfill \\ \hfill z_7& =x_7{y}_1+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{x}_8{y}_2+{\gamma }_{35}{x}_5{y}_3+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{x}_6{y}_4\hfill \\ \hfill & +{\gamma }_{35}{x}_3{y}_5+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{x}_4{y}_6+x_1{y}_7+{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{x}_2{y}_8,\hfill \\ \hfill z_8& =x_8{y}_1+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{x}_7{y}_2+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{x}_6{y}_3+{\gamma }_{45}{x}_5{y}_4\hfill \\ \hfill & +{\gamma }_{45}{x}_4{y}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{x}_3{y}_6+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{x}_2{y}_7+x_1{y}_8.\hfill \end{array}$$

Consider the entries of the matrix product $A\left(X\right)A\left(Y\right)=B$, where $X=(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)$ and $Y=(y_1,y_2,y_3,y_4,y_5,y_6,y_7,y_8)$.

$$\begin{array}{cc}\hfill b_{11}& =(x_1,{\gamma }_{23}{\gamma }_{24}{x}_2,{\gamma }_{23}{\gamma }_{34}{x}_3,{\gamma }_{24}{\gamma }_{34}{x}_4,\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8)\hfill \\ \hfill & ·(y_1,y_2,y_3,y_4,\hfill \\ \hfill & y_5,y_6,y_7,y_8)=z_1,\hfill \\ \hfill b_{12}& =(x_1,{\gamma }_{23}{\gamma }_{24}{x}_2,{\gamma }_{23}{\gamma }_{34}{x}_3,{\gamma }_{24}{\gamma }_{34}{x}_4,\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8)\hfill \\ \hfill & ·({\gamma }_{23}{\gamma }_{24}{y}_2,y_1,{\gamma }_{24}{y}_4,{\gamma }_{23}{y}_3,\hfill \\ \hfill & {\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{y}_6,{\gamma }_{25}{y}_5,{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{y}_8,{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{y}_7)\hfill \\ \hfill & =({\gamma }_{23}{\gamma }_{24}{y}_2{x}_1+{\gamma }_{23}{\gamma }_{24}{y}_1{x}_2+{\gamma }_{23}{\gamma }_{34}{\gamma }_{24}{y}_4{x}_3+{\gamma }_{24}{\gamma }_{34}{\gamma }_{23}{y}_3{x}_4\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{y}_6{x}_5-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{\gamma }_{25}{y}_5{x}_6\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{y}_8{x}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{y}_7{x}_8)\hfill \\ \hfill & ={\gamma }_{23}{\gamma }_{24}{z}_2,\hfill \\ \hfill b_{13}& =(x_1,{\gamma }_{23}{\gamma }_{24}{x}_2,{\gamma }_{23}{\gamma }_{34}{x}_3,{\gamma }_{24}{\gamma }_{34}{x}_4,\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8)\hfill \\ \hfill & ·({\gamma }_{23}{\gamma }_{34}{y}_3,{\gamma }_{34}{y}_4,y_1,{\gamma }_{23}{y}_2,\hfill \\ \hfill & {\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{y}_7,{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{y}_8,{\gamma }_{35}{y}_5,{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{y}_6)\hfill \\ \hfill & =({\gamma }_{23}{\gamma }_{34}{y}_3{x}_1+{\gamma }_{23}{\gamma }_{24}{\gamma }_{34}{y}_4{x}_2+{\gamma }_{23}{\gamma }_{34}{y}_1{x}_3+{\gamma }_{24}{\gamma }_{34}{\gamma }_{23}{y}_2{x}_4\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{y}_7{x}_5-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{y}_8{x}_6\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{y}_8{x}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{y}_6{x}_8)\hfill \\ \hfill & ={\gamma }_{23}{\gamma }_{34}{z}_3,\left[4\right]\hfill \\ \hfill b_{14}& =(x_1,{\gamma }_{23}{\gamma }_{24}{x}_2,{\gamma }_{23}{\gamma }_{34}{x}_3,{\gamma }_{24}{\gamma }_{34}{x}_4,\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8)\hfill \\ \hfill & ·({\gamma }_{24}{\gamma }_{34}{y}_4,{\gamma }_{34}{y}_3,{\gamma }_{24}{y}_2,y_1,\hfill \\ \hfill & {\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{y}_8,{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{y}_7,{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{y}_6,{\gamma }_{45}{y}_5)\hfill \\ \hfill & =({\gamma }_{24}{\gamma }_{34}{y}_4{x}_1+{\gamma }_{23}{\gamma }_{24}{\gamma }_{34}{y}_3{x}_2+{\gamma }_{23}{\gamma }_{24}{\gamma }_{34}{y}_2{x}_3+{\gamma }_{24}{\gamma }_{34}{y}_1{x}_4\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{y}_8{x}_5-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{y}_7{x}_6\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{y}_6{x}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{\gamma }_{45}{y}_5{x}_8)\hfill \\ \hfill & ={\gamma }_{24}{\gamma }_{34}{z}_4,\hfill \\ \hfill b_{15}& =(x_1,{\gamma }_{23}{\gamma }_{24}{x}_2,{\gamma }_{23}{\gamma }_{34}{x}_3,{\gamma }_{24}{\gamma }_{34}{x}_4,\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8)\hfill \\ \hfill & ·(-{\gamma }_{25}{\gamma }_{56}{y}_5,-{\gamma }_{56}{y}_6,-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{y}_7,-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{y}_8,\hfill \\ \hfill & y_1,{\gamma }_{25}{y}_2,{\gamma }_{35}{y}_3,{\gamma }_{45}{y}_4)\hfill \\ \hfill & =(-{\gamma }_{25}{\gamma }_{56}{y}_5{x}_1-{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}{y}_6{x}_2-{\gamma }_{23}{\gamma }_{34}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{y}_7{x}_3-{\gamma }_{24}{\gamma }_{34}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{y}_8{x}_4\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{y}_1{x}_5-{\gamma }_{25}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{y}_2{x}_6\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{\gamma }_{35}{y}_3{x}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{\gamma }_{45}{y}_4{x}_8)\hfill \\ \hfill & =-{\gamma }_{25}{\gamma }_{56}{z}_5,\left[4\right]\hfill \\ \hfill b_{16}& =(x_1,{\gamma }_{23}{\gamma }_{24}{x}_2,{\gamma }_{23}{\gamma }_{34}{x}_3,{\gamma }_{24}{\gamma }_{34}{x}_4,\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8)\hfill \\ \hfill & ·(-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{\gamma }_{56}{y}_6,-{\gamma }_{56}{y}_5,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{y}_8,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{y}_7,\hfill \\ \hfill & {\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{y}_2,y_1,{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{y}_4,{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{y}_3)\hfill \\ \hfill & =(-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{y}_6{x}_1-{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}{y}_5{x}_2-{\gamma }_{23}{\gamma }_{34}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{y}_8{x}_3-{\gamma }_{24}{\gamma }_{34}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{y}_7{x}_4\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}}{{\gamma }_{25}}}{y}_2{x}_5-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{y}_1{x}_6\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{25}}}{y}_4{x}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{25}}}{y}_3{x}_8)\hfill \\ \hfill & =-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{z}_6,\hfill \\ \hfill b_{17}& =(x_1,{\gamma }_{23}{\gamma }_{24}{x}_2,{\gamma }_{23}{\gamma }_{34}{x}_3,{\gamma }_{24}{\gamma }_{34}{x}_4,\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8)\hfill \\ \hfill & ·(-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{y}_7,-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{y}_8,-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{y}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{y}_6,\hfill \\ \hfill & {\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{y}_3,{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{y}_4,y_1,{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{y}_2)\hfill \\ \hfill & =(-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{y}_7{x}_1-{\gamma }_{23}{\gamma }_{24}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{y}_8{x}_2\hfill \\ \hfill & -{\gamma }_{23}{\gamma }_{34}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}{y}_5{x}_3-{\gamma }_{24}{\gamma }_{34}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}{y}_6{x}_4\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{34}}{{\gamma }_{35}}}{y}_3{x}_5-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{35}}}{y}_4{x}_6\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{y}_1{x}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{\displaystyle \frac{{\gamma }_{23}{\gamma }_{45}}{{\gamma }_{35}}}{y}_2{x}_8)\hfill \\ \hfill & =-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{z}_7,\left[4\right]\hfill \\ \hfill b_{18}& =(x_1,{\gamma }_{23}{\gamma }_{24}{x}_2,{\gamma }_{23}{\gamma }_{34}{x}_3,{\gamma }_{24}{\gamma }_{34}{x}_4,\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{x}_5,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{x}_6,-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{x}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{x}_8)\hfill \\ \hfill & ·(-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{y}_8,-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{y}_7,-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{y}_6,-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{y}_5,\hfill \\ \hfill & {\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{y}_4,{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{y}_3,{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{y}_2,y_1)\hfill \\ \hfill & =(-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{y}_8{x}_1-{\gamma }_{23}{\gamma }_{24}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}{y}_7{x}_2\hfill \\ \hfill & -{\gamma }_{23}{\gamma }_{34}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}{y}_6{x}_3-{\gamma }_{24}{\gamma }_{34}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}{y}_5{x}_4\hfill \\ \hfill & -{\gamma }_{25}{\gamma }_{56}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{34}}{{\gamma }_{45}}}{y}_4{x}_5-{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}}{{\gamma }_{45}}}{y}_3{x}_6\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}^2}}{\displaystyle \frac{{\gamma }_{24}{\gamma }_{35}}{{\gamma }_{45}}}{y}_2{x}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{y}_1{x}_8)\hfill \\ \hfill & =-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{45}^2}}{z}_7.\hfill \end{array}$$

We have thus shown that the first row of B coincides with the first row of $A\left(XY\right)$. Computing the remaining entries of B in the same way shows that B coincides with $A\left(XY\right)$. Property 3 is proved.

Property 4 is likewise verified by direct calculation, as noted above. The calculations were performed using symbolic computation in Python.

Property 5 follows immediately from Properties 1 and 4. Property 6 follows from Properties 2 and 4, and Property 7 follows from Properties 3 and 4.

This completes the proof of Theorem 3. □

The theorem establishes an isomorphism between each space of eight-dimensional numbers and the corresponding matrix space; this isomorphism preserves addition and multiplication. Hence, one may work either in the spaces of eight-dimensional numbers or, equivalently, in the associated matrix spaces. Matrix spaces, however, have properties not ordinarily attributed to numbers; in particular, a matrix has eigenvalues. This observation gives rise to additional concepts for eight-dimensional numbers, including their spectrum.

Corollary 1. 

$\left|x\right|=\sqrt[8]{detA\left(x\right)}$.

This corollary provides an alternative definition of the symplectic modulus of an eight-dimensional number.

6. Degenerate Eight-Dimensional Numbers

Definition. An eight-dimensional number is called degenerate if its symplectic modulus is zero.

We study the solutions of ${\left|x\right|}_i=0,i=1,2,\dots ,112$, that is, the degenerate numbers. Explicit solutions of this equation make it possible to describe the general structure of degenerate numbers in all spaces $M_i\left({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}\right),i=1,2,\dots ,122$ in which the generating numbers are strictly positive.

Consider the space $M_1\left({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}\right)$, whose modulus is defined by (28).

It follows from this formula that $M_1\left({\gamma }_{23},{\gamma }_{24},{\gamma }_{25},{\gamma }_{34},{\gamma }_{35},{\gamma }_{45},{\gamma }_{56}\right)$ contains four types of degenerate numbers. Degenerate numbers of the first type are defined by

$$\left\{\begin{array}{c}{x}_1=\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\\ x_5={\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{24}}}{{\gamma }_{25}}}{x}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{34}}}{{\gamma }_{35}}}{x}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{34}}}{{\gamma }_{45}}}{x}_8\end{array}\right.,$$

(29)

Degenerate numbers of the second type are defined by

$$\left\{\begin{array}{c}{x}_1=\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3+\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\\ x_5={\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{24}}}{{\gamma }_{25}}}{x}_6-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{34}}}{{\gamma }_{35}}}{x}_7+{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{34}}}{{\gamma }_{45}}}{x}_8\end{array}\right.,$$

(30)

Degenerate numbers of the third type are defined by

$$\left\{\begin{array}{c}{x}_1=-\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3+\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\\ x_5=-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{24}}}{{\gamma }_{25}}}{x}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{34}}}{{\gamma }_{35}}}{x}_7+{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{34}}}{{\gamma }_{45}}}{x}_8\end{array}\right.,$$

(31)

Degenerate numbers of the fourth type are defined by

$$\left\{\begin{array}{c}{x}_1=-\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\\ x_5=-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{24}}}{{\gamma }_{25}}}{x}_6-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{34}}}{{\gamma }_{35}}}{x}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{34}}}{{\gamma }_{45}}}{x}_8\end{array}\right.,$$

(32)

The four types of degenerate numbers can be described similarly for every other space listed in Appendix A. Thus, the degenerate numbers form four six-dimensional manifolds in the eight-dimensional space.

In the isotropic space $M_1$, the degenerate numbers have the forms

$$\left(x_2+x_3-x_4,x_2,x_3,x_4,x_6+x_7-x_8,x_6,x_7,x_8\right),$$

$$\left(x_2-x_3+x_4,x_2,x_3,x_4,x_6-x_7+x_8,x_6,x_7,x_8\right),$$

$$\left(-x_2+x_3+x_4,x_2,x_3,x_4,-x_6+x_7+x_8,x_6,x_7,x_8\right),$$

$$\left(-x_2-x_3-x_4,x_2,x_3,x_4,-x_6-x_7-x_8,x_6,x_7,x_8\right)$$

for the first, second, third, and fourth types, respectively.

Denote these manifolds by $O_1,O_2,O_3,O_4$, respectively, and introduce the sets $O_{ij}=O_i\cap O_j,i,j=1,2,3,4,i\ne j$ and $O_{ijk}=O_i\cap O_j\cap O_k,i,j,k=1,2,3,4,i\ne j\ne k$.

Theorem 4. 

Degenerate numbers have the following properties:

1. 

Let $x,y\in O_i,i=1,2,3,4$. Then $x+y,x-y\in O_i$.

2. 

Let $x\in O_i,i=1,2,3,4$ and $y\in R^8$. Then $x·y\in O_i$.

3. 

Let $x\in O_i,i=1,2,3,4$ and $y\in O_j,j=1,2,3,4$. Then $x·y\in O_{ij}$.

4. 

Let $x\in O_{ij},i,j=1,2,3,4$ and $y\in O_k,k=1,2,3,4$. Then $x·y\in O_{ijk}$.

5. 

Let $x,y,z,t$ be degenerate numbers of distinct types. Then $x·y·z·t=0$, where 0=(0,0,0,0,0,0,0,0).

Proof. 

Property 1 is immediate.

We prove Property 2. It suffices to consider $M_1$, since the other spaces are treated similarly. First let $x\in O_1$ and put $z=x·y$. The components of z are given by the formulas used in the proof of Theorem 3. We calculate $t=\sqrt{{\gamma }_{23}{\gamma }_{24}}{z}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{z}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{z}_4$.

$$\begin{array}{cc}\hfill t& =\left(\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)y_1\hfill \\ \hfill & +\left(\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_1+{\gamma }_{24}\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_4-{\gamma }_{23}\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_3\right)y_2\hfill \\ \hfill & +\left({\gamma }_{34}\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_4+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_1-{\gamma }_{23}\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_2\right)y_3\hfill \\ \hfill & +\left({\gamma }_{34}\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_3+{\gamma }_{24}\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_2-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_1\right)y_4\hfill \\ \hfill & +\left(-{\gamma }_{56}\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_6-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_7+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_8\right)y_5\hfill \\ \hfill & +\left(-{\gamma }_{56}\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_5-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_8+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_7\right)y_6\hfill \\ \hfill & +\left(-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_8-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{35}}}\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_5+{\displaystyle \frac{{\gamma }_{23}{\gamma }_{56}}{{\gamma }_{35}}}\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_6\right)y_7\hfill \\ \hfill & +\left(-{\displaystyle \frac{{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}{{\gamma }_{35}{\gamma }_{45}}}\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_7-{\displaystyle \frac{{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{45}}}\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_6+{\displaystyle \frac{{\gamma }_{25}{\gamma }_{56}}{{\gamma }_{45}}}\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_5\right)y_8.\hfill \end{array}$$

The factor in parentheses preceding $y_1$ is $x_1$. From the factor in parentheses preceding $y_5$, factor out $-{\gamma }_{25}{\gamma }_{56}$; by (29), the remaining factor is $x_5$. Next, in the factor preceding $y_2,y_3,y_4$, replace $x_1$ by its expression from (29), and in the factor preceding $y_6,y_7,y_8$, replace $x_5$ by its expression from (29). Elementary simplification then gives $t=z_1$.

Calculating $t={\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{24}}}{{\gamma }_{25}}}{z}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{34}}}{{\gamma }_{35}}}{z}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{34}}}{{\gamma }_{45}}}{z}_8$ in the same manner gives $t=z_5$. Thus, the components of z satisfy (29), and z is a degenerate number of the first type.

The cases $x\in O_i,i=2,3,4$ are treated analogously. Property 2 is proved.

Property 3 follows from Property 2. Indeed, if $x\in O_i$ and $y\in O_j$, then Property 2 implies that $x·y$ belongs to both $O_i$ and $O_j$, and therefore belongs to their intersection. Property 4 is proved analogously to Property 2, while Property 5 follows by direct verification.

This completes the proof of Theorem 4. □

Theorem 4 shows that the sets $O_i,i=1,2,3,4$ are closed under addition and multiplication and are therefore subspaces of the space of eight-dimensional numbers. Their intersections are likewise subspaces.

Corollary 2. 

The subspace $O_1\cap O_2\cap O_3\cap O_4$ consists only of the zero element.

Consequently, the four six-dimensional subspaces have a single common point in the eight-dimensional space; equivalently, any two of the corresponding four-dimensional intersections meet only at zero.

We examine the structure of the four-dimensional subspaces $O_{ij},i\ne j$ in $M_1$. The situation in the remaining spaces is entirely analogous. There are six such subspaces: $O_{12},O_{13},O_{14},O_{23},O_{24},O_{34}$. Their degenerate numbers have the following respective forms:

$$\left(\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2,x_2,\sqrt{{\displaystyle \frac{{\gamma }_{24}}{{\gamma }_{23}}}}{x}_4,x_4,{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{24}}}{{\gamma }_{25}}}{x}_6,x_6,\sqrt{{\displaystyle \frac{{\gamma }_{24}}{{\gamma }_{23}}}}{\displaystyle \frac{{\gamma }_{35}}{{\gamma }_{45}}}{x}_8,x_8\right),$$

$$\left(\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3,\sqrt{{\displaystyle \frac{{\gamma }_{34}}{{\gamma }_{23}}}}{x}_4,x_3,x_4,{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{34}}}{{\gamma }_{35}}}{x}_7,\sqrt{{\displaystyle \frac{{\gamma }_{34}}{{\gamma }_{23}}}}{\displaystyle \frac{{\gamma }_{25}}{{\gamma }_{45}}}{x}_8,x_7,x_8\right),$$

$$\left(-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4,-\sqrt{{\displaystyle \frac{{\gamma }_{34}}{{\gamma }_{24}}}}{x}_3,x_3,x_4,-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{34}}}{{\gamma }_{45}}}{x}_7,-\sqrt{{\displaystyle \frac{{\gamma }_{34}}{{\gamma }_{24}}}}{\displaystyle \frac{{\gamma }_{25}}{{\gamma }_{35}}}{x}_8,x_7,x_8\right),$$

$$\left(\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4,\sqrt{{\displaystyle \frac{{\gamma }_{34}}{{\gamma }_{24}}}}{x}_3,x_3,x_4,{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{34}}}{{\gamma }_{45}}}{x}_7,\sqrt{{\displaystyle \frac{{\gamma }_{34}}{{\gamma }_{24}}}}{\displaystyle \frac{{\gamma }_{25}}{{\gamma }_{35}}}{x}_8,x_7,x_8\right),$$

$$\left(-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3,-\sqrt{{\displaystyle \frac{{\gamma }_{34}}{{\gamma }_{23}}}}{x}_4,x_3,x_4,-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{34}}}{{\gamma }_{35}}}{x}_7,-\sqrt{{\displaystyle \frac{{\gamma }_{34}}{{\gamma }_{23}}}}{\displaystyle \frac{{\gamma }_{25}}{{\gamma }_{45}}}{x}_8,x_7,x_8\right),$$

$$\left(-\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2,x_2,-\sqrt{{\displaystyle \frac{{\gamma }_{24}}{{\gamma }_{23}}}}{x}_4,x_4,-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{24}}}{{\gamma }_{25}}}{x}_6,x_6,-\sqrt{{\displaystyle \frac{{\gamma }_{24}}{{\gamma }_{23}}}}{\displaystyle \frac{{\gamma }_{35}}{{\gamma }_{45}}}{x}_8,x_8\right).$$

The general forms of the degenerate numbers in $O_{123},O_{124},O_{134},O_{234}$ can be written similarly.

Lemma 1 

(Zero Divisors). The equation $x·y=0$ in an eight-dimensional space has infinitely many nonzero solutions, each of one of the following forms:

1. 

$x\in O_{12},y\in O_{34}$,

2. 

$x\in O_{13},y\in O_{24}$,

3. 

$x\in O_{14},y\in O_{23}$,

4. 

$x\in O_1,y\in O_{234}$,

5. 

$x\in O_2,y\in O_{134}$,

6. 

$x\in O_3,y\in O_{124}$,

7. 

$x\in O_4,y\in O_{123}$.

The proof follows immediately from Theorem 4.

Consider the equation $x·y=J_1$, where $J_1$ is the first basis number and x is a nondegenerate eight-dimensional number. Multiply both sides by the conjugate $x^{*}$ of x: $x^{*}·x·y=x^{*}$. By the definition of conjugation, $x^{*}·x={\left|x\right|}^8{J}_1$. Hence ${\left|x\right|}^{8}y=x^{*}$, and division by ${\left|x\right|}^8$ gives

$$y={\displaystyle \frac{1}{{\left|x\right|}^8}}{x}^{*}.$$

(33)

The resulting number y is called the reciprocal of x and is denoted by $x^{-1}$. Formula (13) shows that only nondegenerate numbers have reciprocals. Division by a nondegenerate number can therefore be defined by ${\displaystyle \frac{x}{y}}=x·y^{-1}$, provided that y is nondegenerate.

Lemma 2. 

If x is a nondegenerate eight-dimensional number, then $A\left(x^{-1}\right)=A^{-1}\left(X\right)$.

Proof. 

The identity $x^{-1}={\displaystyle \frac{1}{{\left|x\right|}^8}}{x}^{*}$ implies $A\left(x^{-1}\right)={\displaystyle \frac{1}{{\left|x\right|}^8}}A\left(x^{*}\right)$. Multiplying both sides by $A\left(x\right)$, we obtain $A\left(x^{-1}\right)A\left(x\right)={\displaystyle \frac{1}{{\left|x\right|}^8}}A\left(x^{*}\right)A\left(x\right)=A\left({\displaystyle \frac{1}{{\left|x\right|}^8}}{x}^{*}x\right)=A\left(J_1\right)$, where Theorem 2 has been used. The matrix $A\left(J_1\right)$ is the identity matrix E; therefore, $A\left(x^{-1}\right)=A^{-1}\left(X\right)$.

This proves Lemma 2. □

It follows immediately from the lemma that the matrix $A^{-1}\left(X\right)$ has one of the forms (16)-(20).

7. Spectrum of Eight-Dimensional Numbers

Definition. The spectrum of an eight-dimensional number x is the collection of eigenvalues of the corresponding matrix $A\left(x\right)$.

We write the characteristic equation for the spectrum of an eight-dimensional number $x=(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)$ in $M_1$; the argument is analogous for the remaining spaces: $A\left(x\right)-\mu x=0$.

The determinant of $A-\mu E$ coincides with the determinant of A after $x_1$ is replaced by $x_1-\mu$. Hence, the characteristic equation is

$$\begin{array}{c}det\left(A\left(X\right)-\mu E\right)=[{\left(x_1-\mu -\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3+\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)}^2\hfill \\ +{\left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5-\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7+{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)}^2]\\ \times [{\left(x_1-\mu -\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)}^2\\ +{\left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5-\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)}^2]\\ \times [{\left(x_1-\mu +\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)}^2\\ +{\left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5+\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)}^2]\\ \times [{\left(x_1-\mu +\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3+\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\right)}^2\\ \hfill +{\left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5+\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7+{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)}^2]=0.\end{array}$$

The eigenvalues obtained from this equation are

$$\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}{\mu }_{1,2}=x_1-\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3+\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\pm \\ \left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5-\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7+{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)i\end{array}\\ \begin{array}{c}{\mu }_{3,4}=x_1-\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\pm \\ \left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5-\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)i\end{array}\end{array}\\ \begin{array}{c}\begin{array}{c}{\mu }_{5,6}=x_1+\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2-\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3-\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\pm \\ \left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5+\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6-{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7-{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)i\end{array}\\ \begin{array}{c}{\mu }_{7,8}=x_1+\sqrt{{\gamma }_{23}{\gamma }_{24}}{x}_2+\sqrt{{\gamma }_{23}{\gamma }_{34}}{x}_3+\sqrt{{\gamma }_{24}{\gamma }_{34}}{x}_4\pm \\ \left(\sqrt{{\gamma }_{25}{\gamma }_{56}}{x}_5+\sqrt{{\displaystyle \frac{{\gamma }_{23}{\gamma }_{24}{\gamma }_{56}}{{\gamma }_{25}}}}{x}_6+{\displaystyle \frac{\sqrt{{\gamma }_{23}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{35}}}{x}_7+{\displaystyle \frac{\sqrt{{\gamma }_{24}{\gamma }_{25}{\gamma }_{34}{\gamma }_{56}}}{{\gamma }_{45}}}{x}_8\right)i\end{array}\end{array}\end{array}\right..$$

(34)

Thus, the spectrum of an eight-dimensional number in $M_1$ consists of the eight complex numbers in (34), arranged in conjugate pairs. In the isotropic space $M_1$, the spectrum is

$$\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}{\mu }_1=x_1-x_2-x_3+x_4+\left(x_5-x_6-x_7+x_8\right)i\\ {\mu }_2=x_1-x_2-x_3+x_4-\left(x_5-x_6-x_7+x_8\right)i\end{array}\\ \begin{array}{c}{\mu }_3=x_1-x_2+x_3-x_4+\left(x_5-x_6+x_7-x_8\right)i\\ {\mu }_4=x_1-x_2+x_3-x_4-\left(x_5-x_6+x_7-x_8\right)i\end{array}\end{array}\\ \begin{array}{c}\begin{array}{c}{\mu }_5=x_1+x_2-x_3-x_4+\left(x_5+x_6-x_7-x_8\right)i\\ {\mu }_6=x_1+x_2-x_3-x_4-\left(x_5+x_6-x_7-x_8\right)i\end{array}\\ \begin{array}{c}{\mu }_7=x_1+x_2+x_3+x_4+\left(x_5+x_6+x_7+x_8\right)i\\ {\mu }_8=x_1+x_2+x_3+x_4-\left(x_5+x_6+x_7+x_8\right)i\end{array}\end{array}\end{array}\right..$$

(35)

Denote the spectrum of x by $\Lambda \left(x\right)$ and consider the mapping $S:x\to \Lambda \left(x\right)$ on the spaces $M_i,i=1,2,3,\dots ,122$.

Theorem 5. 

The mapping S is one-to-one and onto; hence, it is a bijection.

Proof. It suffices to consider $M_1$, since the proof for the remaining spaces is analogous. For simplicity, we consider the isotropic space $M_1$.

Suppose that $x\ne y$. We show that $\Lambda \left(x\right)\ne \Lambda \left(y\right)$. If not, then ${\mu }_i\left(x\right)={\mu }_i\left(y\right),i=1,2,3,4,5,6,7,8$, and therefore

$$\left\{\begin{array}{cc}& x_1-x_2-x_3+x_4+(x_5-x_6-x_7+x_8)i\hfill \\ & =y_1-y_2-y_3+y_4+(y_5-y_6-y_7+y_8)i,\hfill \\ & x_1-x_2-x_3+x_4-(x_5-x_6-x_7+x_8)i\hfill \\ & =y_1-y_2-y_3+y_4-(y_5-y_6-y_7+y_8)i,\hfill \\ & x_1-x_2+x_3-x_4+(x_5-x_6+x_7-x_8)i\hfill \\ & =y_1-y_2+y_3-y_4+(y_5-y_6+y_7-y_8)i,\hfill \\ & x_1-x_2+x_3-x_4-(x_5-x_6+x_7-x_8)i\hfill \\ & =y_1-y_2+y_3-y_4-(y_5-y_6+y_7-y_8)i,\hfill \\ & x_1+x_2-x_3-x_4+(x_5+x_6-x_7-x_8)i\hfill \\ & =y_1+y_2-y_3-y_4+(y_5+y_6-y_7-y_8)i,\hfill \\ & x_1+x_2-x_3-x_4-(x_5+x_6-x_7-x_8)i\hfill \\ & =y_1+y_2-y_3-y_4-(y_5+y_6-y_7-y_8)i,\hfill \\ & x_1+x_2+x_3+x_4+(x_5+x_6+x_7+x_8)i\hfill \\ & =y_1+y_2+y_3+y_4+(y_5+y_6+y_7+y_8)i,\hfill \\ & x_1+x_2+x_3+x_4-(x_5+x_6+x_7+x_8)i\hfill \\ & =y_1+y_2+y_3+y_4-(y_5+y_6+y_7+y_8)i.\hfill \end{array}\right.$$

Equating real and imaginary parts and simplifying gives $x=y$, a contradiction.

Conversely, we show that every spectrum consisting of numbers of the form (35) corresponds to exactly one eight-dimensional number. Indeed, let

$${\mu }_{1,2}=a_1\pm b_{1}i,$$

$${\mu }_{3,4}=a_2\pm b_{2}i,$$

$${\mu }_{5,6}=a_3\pm b_{3}i,$$

$${\mu }_{7,8}=a_4\pm b_{4}i,$$

be the spectrum of an eight-dimensional number. Formulas (35) then imply

$$a_1=x_1-x_2-x_3+x_4,$$

$$b_1=x_5-x_6-x_7+x_8,$$

$$a_2=x_1-x_2+x_3-x_4,$$

$$b_2=x_5-x_6+x_7-x_8,$$

$$a_3=x_1+x_2-x_3-x_4,$$

$$b_3=x_5+x_6-x_7-x_8,$$

$$a_4=x_1+x_2+x_3+x_4,$$

$$b_4=x_5+x_6+x_7+x_8,$$

which determine $x_i,i=1,2,\dots ,8$ uniquely. □

Corollary 3. 

The only number with zero spectrum is $0=(0,0,0,0,0,0,0,0)$.

Corollary 4. 

The spectra of the basis numbers in the isotropic space $M_1$ are

$$\begin{array}{cccc}\hfill \Lambda \left(J_1\right)& =(1,1,1,1,1,1,1,1),\hfill & \hfill \Lambda \left(J_2\right)& =(-1,-1,-1,-1,1,1,1,1),\hfill \\ \hfill \Lambda \left(J_3\right)& =(-1,-1,1,1,-1,-1,1,1),\hfill & \hfill \Lambda \left(J_4\right)& =(1,1,-1,-1,-1,-1,1,1),\hfill \\ \hfill \Lambda \left(J_5\right)& =(\mathrm{i},-\mathrm{i},\mathrm{i},-\mathrm{i},\mathrm{i},-\mathrm{i},\mathrm{i},-\mathrm{i}),\hfill & \hfill \Lambda \left(J_6\right)& =(-\mathrm{i},\mathrm{i},-\mathrm{i},\mathrm{i},\mathrm{i},-\mathrm{i},\mathrm{i},-\mathrm{i}),\hfill \\ \hfill \Lambda \left(J_7\right)& =(-\mathrm{i},\mathrm{i},\mathrm{i},-\mathrm{i},-\mathrm{i},\mathrm{i},\mathrm{i},-\mathrm{i}),\hfill & \hfill \Lambda \left(J_8\right)& =(\mathrm{i},-\mathrm{i},-\mathrm{i},\mathrm{i},-\mathrm{i},\mathrm{i},\mathrm{i},-\mathrm{i}).\hfill \end{array}$$

Lemma 3. 

Every eight-dimensional number x satisfies

$${\left|x\right|}^8={\mu }_1·{\mu }_2·{\mu }_3·{\mu }_4·{\mu }_5·{\mu }_6·{\mu }_7·{\mu }_8.$$

(36)

The proof is immediate.

Theorem 6. 

The spectrum of an eight-dimensional number has the following properties:

1. 

${\mu }_i\left(x\pm y\right)={\mu }_i\left(x\right)\pm {\mu }_i\left(y\right)$ for all $x\in R^8$, $y\in R^8$, and $i=1,2,\dots ,8$;

2. 

${\mu }_i\left(x·y\right)={\mu }_i\left(x\right)·{\mu }_i\left(y\right)$ for all $x\in R^8$, $y\in R^8$, and $i=1,2,\dots ,8$;

3. 

${\mu }_i\left(b·x\right)=b·{\mu }_i\left(x\right)$ for all $x\in R^8$, $b\in R^1$, and $i=1,2,\dots ,8$;

4. 

${\mu }_i\left(x^{-1}\right)={\left({\mu }_i\left(x\right)\right)}^{-1}$ for every nondegenerate $x\in R^8$ and $i=1,2,\dots ,8$;

5. 

${\mu }_i\left(x^n\right)={\mu }_i^n\left(x\right)$ for all $x\in R^8$, $i=1,2,\dots ,8$, and $n\in N$;

where ${\mu }_i\left(x\right)$ is the ith component of the spectrum of x.

Proof. For simplicity, we prove the assertion only for the isotropic space $M_1$; the other spaces are treated analogously.

The first property is immediate. We prove the second. Put $z=x·y$. Then

$$z_1=x_1{y}_1+x_2{y}_2+x_3{y}_3+x_4{y}_4-x_5{y}_5-x_6{y}_6-x_7{y}_7-x_8{y}_8,$$

$$z_2=x_2{y}_1+x_1{y}_2+x_4{y}_3+x_3{y}_4-x_6{y}_5-x_5{y}_6-x_8{y}_7-x_7{y}_8,$$

$$z_3=x_3{y}_1+x_4{y}_2+x_1{y}_3+x_2{y}_4-x_7{y}_5-x_8{y}_6-x_5{y}_7-x_6{y}_8,$$

$$z_4=x_4{y}_1+x_3{y}_2+x_2{y}_3+x_1{y}_4-x_8{y}_5-x_7{y}_6-x_6{y}_7-x_5{y}_8,$$

$$z_5=x_5{y}_1+x_6{y}_2+x_7{y}_3+x_8{y}_4+x_1{y}_5+x_2{y}_6+x_3{y}_7+x_4{y}_8,$$

$$z_6=x_6{y}_1+x_5{y}_2+x_8{y}_3+x_7{y}_4+x_2{y}_5+x_1{y}_6+x_4{y}_7+x_3{y}_8,$$

$$z_7=x_7{y}_1+x_8{y}_2+x_5{y}_3+x_6{y}_4+x_3{y}_5+x_4{y}_6+x_1{y}_7+x_2{y}_8,$$

$$z_8=x_8{y}_1+x_7{y}_2+x_6{y}_3+x_5{y}_4+x_4{y}_5+x_3{y}_6+x_2{y}_7+x_1{y}_8.$$

$$\begin{array}{cc}\hfill {\mu }_1\left(z\right)=& \left(x_1-x_2-x_3+x_4\right)y_1+\left(-x_1+x_2+x_3-x_4\right)y_2\hfill \\ \hfill & +\left(-x_1+x_2+x_3-x_4\right)y_3+\left(x_1-x_2-x_3+x_4\right)y_4\hfill \\ \hfill & -\left(x_5-x_6-x_7+x_8\right)y_5-\left(-x_5+x_6+x_7-x_8\right)y_6\hfill \\ \hfill & -\left(-x_5+x_6+x_7-x_8\right)y_7-\left({-x}_5+x_6+x_7-x_8\right)y_8\hfill \\ \hfill & +i(\left(x_5-x_6-x_7+x_8\right)y_1+\left(-x_5+x_6+x_7-x_8\right)y_2\hfill \\ \hfill & +\left(-x_5+x_6+x_7-x_8\right)y_3+\left(x_5-x_6-x_7+x_8\right)y_4\hfill \\ \hfill & +\left(x_1-x_2-x_3+x_4\right)y_5+\left(-x_1+x_2+x_3-x_4\right)y_6\hfill \\ \hfill & +\left(-x_1+x_2+x_3-x_4\right)y_7+\left(x_1-x_2-x_3+x_4\right)y_8)\hfill \\ \hfill =& \left(x_1-x_2-x_3+x_4\right)\left(y_1-y_2-y_3+y_4\right)\hfill \\ \hfill & +\left(x_5-x_6-x_7+x_8\right)\left(x_5-x_6-x_7+x_8\right)\hfill \\ \hfill & +i(\left(x_1-x_2-x_3+x_4\right)\left(y_5-y_6-y_7+y_8\right)\hfill \\ \hfill & +\left(x_5-x_6-x_7+x_8\right)\left(y_1-y_2-y_3+y_4\right))\hfill \\ \hfill =& {\mu }_1\left(x\right){\mu }_1\left(y\right).\hfill \end{array}$$

The remaining components of the spectrum of z are calculated in the same way.

Properties 3 and 5 follow from Property 2. To prove Property 4, note that Property 2 gives ${\mu }_i\left(x·x^{-1}\right)={\mu }_i\left(x\right)·{\mu }_i\left(x^{-1}\right)$. On the other hand, $x·x^{-1}=J_1$. By the corollary to Theorem 4, ${\mu }_i\left(J_1\right)=1$ for every $i=1,2,\dots ,8$. Therefore, ${\mu }_i\left(x^{-1}\right)={\displaystyle \frac{1}{{\mu }_i\left(x\right)}}$.

This completes the proof of Theorem 6. □

This theorem shows that the isomorphism S is a homomorphism with respect to addition and multiplication. Consequently, eight-dimensional numbers can be treated not only through their matrix representations but also through their spectra. Passing to spectra is a powerful tool for studying eight-dimensional numbers, just as in the four-dimensional case.