Lines at Sixty Degrees: The Jordan Factors of Matsuo Algebras

1. Introduction

A three-transposition group determines a commutative algebra whose distinguished idempotents record the generating involutions and their products. This Matsuo algebra need not satisfy the Jordan identity. One can therefore ask which relations must be imposed to obtain a Jordan algebra, and what algebra results. We study this question at parameter 1/2. The ideal generated by all failures of the Jordan identity is the least ideal with this property. Determining it identifies the largest Jordan quotient of the Matsuo algebra.

De Medts and Rehren classified the connected Matsuo algebras that are themselves Jordan [9]. Gorshkov, Mamontov and Staroletov then classified the cases with nonzero Frobenius radical and Jordan quotient by that radical [15, Proposition 5.2]. Their list consists of two infinite families and seven exceptional cases. Problem 5.1 of asks for the smallest ideal giving a Jordan quotient and for the corresponding factors. Theorems 3.2 and 3.4 answer these questions: the least ideal is the Frobenius radical, and Table 1 identifies every factor over ℚ. Scalar extension gives the answer over every field of characteristic zero.

The construction comes from line geometry. In a three-system, distinct lines meet at 60^∘ or 90^∘, and reflection in any line preserves the set. Their rank-one orthogonal projectors satisfy the Matsuo multiplication rule at parameter 1/2. The resulting homomorphism carries the Frobenius form to the trace form, so its kernel is exactly the Frobenius radical. Theorem 3.1 proves the converse: if a finite connected Matsuo algebra has Jordan quotient by its Frobenius radical, the images of its axes can be realized by such projectors over ℝ, ℂ or ℍ. The main step is to prove positivity of the quotient form by finite-group averaging in an associative representation.

This real realization does not by itself determine the rational algebra. Section 3.3 retains the Hermitian form in the coordinate construction and descends the projector map to ℚ. Eight cases have standard symmetric or Hermitian matrix factors. The E₆ factor consists of the endomorphisms self-adjoint for ⟨1,1,1,1,1,3⟩; it is not Sym₆(ℚ). This qualifies [9, Remark 3.5(i)]. Appendix B gives the corresponding criterion for the radical-free A_n family.

The projector map also makes the relations in the quotient concrete. Orthogonal line sets spanning the same subspace have the same projector sum. Section 4 determines when these frame relations generate the radical and gives star relations for the remaining family. For S₁, the action on nine frames supplies the irreducibility needed in Theorem 3.2; Appendix A provides an independent certificate using Jordan defects. The spectral and power-associative consequences appear in Corollaries 3.5–3.6. Section 5 places the classified line sets within the classical special bound and projective design theory.

The E₇ case leads to a further construction. Its root lines correspond to the transvections of Sp₆(2), and incidence with quadratic forms describes the quotient. Section 6 extends these incidence maps and radical generators to Sp_2n(2) at two singular parameter values. The quotient modules and forms are explicit, although the quotients for n ≥ 4 are not Jordan. Section 7 works out E₇, and Appendix C relates its incidence coordinates to the classical root, bitangent and equiangular models. Finally, Section 8 passes through finite Fischer subspaces to identify the infinite connected systems with Jordan quotient and their real factors.

2. Projectors and Frobenius Radicals

The Frobenius radical is a natural candidate for the relations sought in the introduction. We first recall the Matsuo construction and then realize this radical as the kernel of a map to self-adjoint operators. Unless specified otherwise, Matsuo algebras and incidence modules are over ℚ, and geometric Jordan algebras are over ℝ; all fields considered have characteristic zero.

A three-transposition group (G,D) is generated by a conjugacy-stable set D of involutions such that each product of two members has order at most three. Its Fischer space has point set X = D and lines {x,y,xyx} for noncommuting x,y ∈ D. Two distinct points are collinear exactly when the corresponding involutions do not commute. The space is connected when this collinearity graph is connected. A Fischer subspace is a subset closed under taking the third point of a line.

For a finite Fischer space X and a parameter η ≠ 0,1, the Matsuo algebra M_η(X) has basis {a_x:x ∈ X}. Its multiplication extends bilinearly from $a_x^2=a_x$, zero products for distinct noncollinear points, and the following rule on a line {x,y,z}:

$$a_x{a}_y={\displaystyle \frac{\eta }{2}}\left(a_x+a_y-a_z\right).$$

(1)

A Frobenius form is a symmetric bilinear form satisfying F(ab,c) = F(a,bc). The normalized Matsuo form has F(a_x,a_x) = 1 and matrix G_η = I + (η/2)A, where A is the collinearity adjacency matrix [32, 18, 19, 9]. We write M^⊥ = radF for the vectors orthogonal to every element of M. The next lemma explains why one can quotient by this subspace even when the form is degenerate.

Lemma 2.1.The radical R of an associating form on a commutative algebra M is an ideal. The form induced on M/R is nondegenerate.

Proof. For r ∈ R and a,b ∈ M, we have F(ar,b) = F(r,ab) = 0. Thus ar ∈ R. Products and form values consequently descend to the quotient, whose form has zero radical by definition. □

A commutative algebra is Jordan if (x²y)x = x²(yx) for all x,y. We write I_J(M) for the ideal generated by the differences of these two expressions, called Jordan defects. Thus M/I_J(M) is Jordan, and I_J(M) is contained in every ideal with Jordan quotient. For a connected Matsuo algebra, every proper ideal lies in the Frobenius radical [27]. It follows that a nonzero Jordan quotient can exist only if M/M^⊥ is Jordan. The least-ideal question asks how much of M^⊥ is needed to impose the Jordan identity.

Orthogonal projectors provide a model for the basis idempotents. Let K be ℝ, ℂ or the quaternions ℍ, and let X be a finite set of distinct one-dimensional right K-subspaces of the positive Hermitian space K^r. We call X a three-system if the absolute inner product of unit representatives of distinct lines is 0 or 1/2 and the reflections s_x = I − 2P_x preserve X, where P_x is orthogonal projection onto x. Thus the angles are 90^∘ and 60^∘. The system is indecomposable if it cannot be partitioned into two nonempty mutually orthogonal subsets. Its reflection Fischer space has third point s_x(y) for nonorthogonal x,y [13, 1, 17]. See [16] for the classification.

Write Herm_r(K) for the real vector space of self-adjoint r × r matrices over K.

Proposition 2.2.For such a three-system, the map

Ψ:M_1/2(X;ℝ) → Herm_r(K),  a_x ↦ P_x

is an algebra homomorphism, where the target has product S∘T = (ST + TS)/2. Its image is the real span of the projections, and

F(a,b) = Retr(Ψ(a)Ψ(b)),  kerΨ = radF.

(2)

In particular, the Frobenius form is positive semidefinite and the quotient is isomorphic to this Jordan subalgebra. For real root systems this is the projector construction of [9,Section 3].

Proof. The idempotent and orthogonal products are immediate. For two nonorthogonal lines, choose unit representatives u,v with u^*v = 1/2 by changing the phase of v. Then u − v is a unit vector on the third line. Hence

$$P_u\circ P_v={\displaystyle \frac{1}{4}}\left(u{v}^{*}+v{u}^{*}\right)={\displaystyle \frac{1}{4}}\left(P_u+P_v-P_{u-v}\right).$$

This is the Matsuo product at η = 1/2, over each of the three associative division algebras K. The real trace pairing of the projections equals 1 on the diagonal, 0 for orthogonal pairs and 1/4 otherwise, proving (2). For a Hermitian matrix T,

$$\mathrm{Retr}\left(T^2\right)=\sum _i{T}_{\mathit{ii}}^2+2\sum _{i<j}{\mid T_{\mathit{ij}}\mid }^2.$$

This is positive unless T = 0. The form identity therefore gives the kernel assertion and positivity. □

For an A₂ triple, Figure 1 shows the passage from lines to projectors. The projector coordinates double the direction angle, turning the three lines into the vertices of an equilateral triangle.

Proposition 2.2 turns linear relations among projectors into defining relations for the quotient.It does not assert that the projectors span the full ambient Hermitian algebra;Section 3determines that image. The trace realization also gives the following form inequality.

Corollary 2.3 (Norton inequality).In every projector quotient above,

$$F\left(x^2,y^2\right)-F\left(\mathit{xy},\mathit{xy}\right)={\displaystyle \frac{1}{4}}\parallel \mathit{XY}-\mathit{YX}{\parallel }^2\ge 0,$$

where X = Ψ(x), Y = Ψ(y) and ∥Z∥² = Retr(Z^*Z). This follows by expanding the two sides and using cyclicity of the real trace. It applies equally to the pullback form on M. See [14, Theorem 6.5] for this inequality in Euclidean Jordan algebras.

3. Classification and the Least Jordan Ideal

We now show that the projector construction accounts for every finite connected Matsuo algebra whose quotient by the Frobenius radical is Jordan. We then determine the least ideal and the rational algebra in each case of the Gorshkov–Mamontov–Staroletov list. Throughout this section the parameter is 1/2. The geometric argument is over ℝ; the least-ideal and factor identifications extend to all fields of characteristic zero.

3.1. Jordan Quotients and Three-Systems

For an idempotent e in a real Jordan algebra J, write J_λ(e) = {z ∈ J:e∘z = λz} for its Peirce eigenspaces. In a Jordan quotient of M_1/2 by an ideal contained in its Frobenius radical, the images of the Matsuo axes have J₁(e) = ℝe. Their Peirce involutions act as −1 on J_1/2(e) and as +1 on J₀(e)⊕J₁(e), and permute the axes as the corresponding Fischer-space involutions [18]. These involutions will be used to obtain a positive definite form.

Theorem 3.1 (geometric realization).Let (G,D) be a finite connected 3-transposition group with ∣D∣ ≥ 2, and put M = M_1/2(G,D) over ℝ. Then J = M/M^⊥ is Jordan if and only if its Fischer space is the Fischer space of a finite indecomposable three-system over ℝ, ℂ or ℍ. In the Jordan case there is a choice of K ∈ {ℝ,ℂ,ℍ} and r ≥ 2 such that

$$J\cong {\mathrm{Herm}}_r\left(K\right), \overline{a_x}\mapsto P_{{\mathcal{l}}_x},$$

with the normalized Frobenius form carried to Retr(XY). The lines 𝓁_x form a spanning indecomposable three-system. In particular the form on M is positive semidefinite. Rank two occurs only for the three-point system A₂.

The difficult direction is to recover a positive line geometry from the Jordan quotient. The proof uses the structure theory of finite-dimensional Jordan algebras and finite-group averaging, without the finite classification of three-transposition groups.

Proof. The implication from a three-system follows from Proposition 2.2. Its projector span is a Jordan algebra; an arbitrary ambient choice of K need not make that span all of Herm_r(K).

Conversely, suppose that J = M/M^⊥ is Jordan. Every proper ideal of M lies in M^⊥ [27], so J is simple. The images e_x of the axes are distinct idempotents with J₁(e_x) = ℝe_x: their pairwise Frobenius products are 0 or 1/4, their squared lengths are one, and quotienting preserves the one-dimensional 1-eigenspace. The centroid consists of the linear maps commuting with all multiplication operators. If γ belongs to the centroid, then γe_x ∈ ℝe_x. Thus γ − c kills a nonzero axis for some c ∈ ℝ; its kernel is an ideal, so γ = c. Hence J is central simple and J_ℂ is simple [23]. Write r for its degree and T for its generic trace. Then T(1) = r and T(e_x) = 1.

Put N = ∣D∣, let κ be the collinearity valency, and set $s=\sum _x{e}_x$. Pairing line partners in the Matsuo product gives e_x∘s = (1 + κ/4)e_x. Since the axes span J, it follows that s = (1 + κ/4)1. Taking traces and using κ ≤ N − 1 gives

N = r(1 + κ/4),  N(4 − r) ≤ 3r.

Degree one is impossible because the axes are distinct. In degree two, N ≤ 3; connectedness gives a Fischer line, so the space is A₂, whose projector quotient is Sym₂(ℝ). An exceptional central simple Jordan algebra has degree 3 and dimension 27 [24]. It would give 27 ≤ N ≤ 9, a contradiction. The same structure theorem therefore makes J special, meaning that it embeds in an associative algebra equipped with the symmetrized product.

To prove positivity, we seek an associative representation in which the axes become orthogonal projectors. The unital special envelope of J is finite-dimensional: for a basis x₁,…,x_d, the relations x_jx_i = 2(x_i∘x_j) − x_ix_j and $x_i^2=x_i\circ x_i$ reduce every word to a linear combination of ordered square-free words. Specialness makes the canonical map injective [24]. The intersection of J with the associative radical is zero, since that intersection is a Jordan ideal and the radical contains no nonzero idempotent. Choose a simple factor of the semisimple quotient on which J has nonzero image. Simplicity makes this representation faithful, and its image generates

A = End_K(V),  K ∈ {ℝ,ℂ,ℍ}.

For an axis e, put g_e = 1 − 2e. The Peirce decomposition in A⁺ gives τ_e = Ad(g_e) on J. These involutions permute the finite spanning set of axes, so they generate a finite group Γ. The group $\tilde{\Gamma }=⟨g_e⟩$ maps onto Γ; its kernel commutes with J, hence with the generated algebra A, and lies in Z(A)^×. Every element of $\tilde{\Gamma }$ has real determinant ±1. Thus every kernel scalar has absolute value one and preserves any positive definite K-Hermitian form h₀ on V.

Choose a lift $\tilde{\gamma }$ for each γ ∈ Γ and set

$$h\left(v,w\right)=\sum _{\gamma \in \Gamma }{h}_0\left(\tilde{\gamma }v,\tilde{\gamma }w\right).$$

This is independent of the chosen lifts and invariant under $\tilde{\Gamma }$. For the real inner product Reh, each g_e is an orthogonal involution and each e is an orthogonal projector. Hence all of J consists of self-adjoint operators, and β(x,y) = tr_ℝ,V(xy) is a positive definite associative form. The comparison operator between β and the nondegenerate Frobenius form F belongs to the centroid. Thus β = cF, with c = β(e,e) > 0, proving positivity of F.

The classification of simple Euclidean Jordan algebras now gives J≅Herm_r(K) for K = ℝ,ℂ or ℍ; degree two and the exceptional case have already been settled.

Finally primitive idempotents in these matrix algebras are rank-one projectors. The normalized trace form agrees with the Frobenius form: both are associative and take value one at an axis. Their trace products give precisely angles 90^∘ and 60^∘. The Matsuo relation and the projector identity show that the third point on a Fischer line is the reflected line. Connectedness gives indecomposability. The projectors span all of J by construction, so the lines span K^r. □

The indecomposable three-systems are A_n,D_n,E₆,E₇,E₈ over ℝ, the additional complex systems G(3,3,m),K₅,K₆, and the additional quaternionic systems S₁,U [6, 7, 30; 16]. Here K₅ and K₆ are also denoted G₃₃ and G₃₄.

3.2. The Nine Cases and the Smallest Jordan Ideal

We now return to the least ideal I_J = I_J(M). Table 1 lists the cases where the Frobenius radical is nonzero and the quotient is Jordan [15, Proposition 5.2]. Rows 1 and 2 have m ≥ 4. Here N is the number of axes, J = M/M^⊥, and h = dimJ_1/2(e) is the dimension of the half-Peirce space in the quotient at an axis e.

The last column anticipates the rational identifications in Theorem 3.4. Put $E=\mathbb{Q}\left(\sqrt{-3}\right)$, $\omega =\left(-1+\sqrt{-3}\right)/2$, ℍ_ℚ = (−1,−1)_ℚ and q₆=⟨1,1,1,1,1,3⟩. The notation Herm_r(E) and Herm_r(ℍ_ℚ) uses the standard adjoint; Herm₆(ℚ,q₆) consists of the endomorphisms self-adjoint for q₆.

Theorem 3.2. In each of the nine cases, I_J = M^⊥ over every characteristic-zero field. Over ℝ, the factors are the real scalar extensions of the algebras in Table 1.

Proof. The ideal I_J is characteristic, is contained in M^⊥ because that quotient is Jordan, and is nonzero because these Matsuo algebras are not Jordan [9]. It therefore suffices to prove that M^⊥ is irreducible under a group of automorphisms.

For D_m, let x_ij be the sum of the two axes associated with e_i + e_j and e_i − e_j, and let y_ij be their difference. The projector images of the y_ij are independent off-diagonal symmetric matrices, whereas x_ij maps to E_ii + E_jj. Consequently

$$M^{\perp }=\left\{\sum _{i<j}{c}_{\mathit{ij}}{x}_{\mathit{ij}}:\sum _{j\ne i}{c}_{\mathit{ij}}=0\mathrm{for\; every}i\right\}.$$

By the two-subset permutation-module decomposition [25], this is the irreducible characteristic-zero 𝔖_m-module S^(m−2,2) inside the permutation module on two-subsets. This proves the assertion uniformly for all m ≥ 4.

For G(3,3,m), let σ_ij be the sum of the three axes on e_i − ω^te_j, t = 0,1,2. Their projector sum is ${\displaystyle \frac{3}{2}}\left(E_{\mathit{ii}}+E_{\mathit{jj}}\right)$; the coefficient-sum-zero combinations for each pair map injectively to its two-dimensional real off-diagonal Hermitian space. The kernel is therefore the same two-subset module, with σ_ij in place of x_ij, and is again S^(m−2,2).

In cases 3–8, the action on the transposition class is primitive of rank three by Fischer’s theorem [13, Theorem 3.3.5; 20, Section 6]. Its orbits on ordered pairs are equality, distinct commuting pairs and noncommuting pairs. Thus its permutation commutant has dimension three. The three adjacency eigenspaces are nonzero invariant summands, so each is absolutely irreducible, including M^⊥. In case 9, the stabilizer action described after Proposition 4.4 is two-transitive on the nine frames. The augmentation module, consisting of frame coefficients with sum zero, is M^⊥, so the same argument applies. Appendix A gives an independent certificate by Jordan defects. This proves I_J = M^⊥ over ℝ.

All multiplication constants are rational. Both the Frobenius kernel and the ideal generated by the polarized Jordan defects commute with characteristic-zero scalar extension. Equality over ℝ therefore gives equality over ℚ, then over every characteristic-zero field. The real factor types follow from the preceding theorem and the displayed pairs (dimJ,h): the equations

dimJ = r+Δr(r − 1)/2,  h=Δ(r − 1)

determine r and Δ∈{1,2,4}. □

The rank-three step above is a consequence of Fischer’s theorem; exact orbit enumeration provides an independent check. In the order (diagonal, distinct commuting, noncommuting), the orbital sizes for cases 3–8 are

(120,7560,6720), (36,540,720), (63,1890,2016),

(126,5670,10080), (45,540,1440), (165,5940,21120).

The dimensions in Table 1 can also be read directly from the multiplication and Frobenius matrices. For an axis a, let L_a be multiplication by a, and put T = 4L_a and G₄ = 4I + A. Then the half-Peirce dimension in the quotient is

h = rank_ℚ(G₄(4T − T²)),

because 4T − T² is four times the half-Peirce projector. The Gram rank gives the quotient dimension, while this formula gives the listed half-Peirce dimensions.

3.3. The Factors Over the Rationals

Identifying the matrix algebra over ℝ leaves its rational Hermitian form undetermined. For a nondegenerate Hermitian space (V,h), write Herm(V,h) for its self-adjoint endomorphisms, with product (XY + YX)/2. The following descent lemma reduces the rational identification to a coordinate model and a proof that its projectors span this algebra.

Lemma 3.3 (projector descent).Let K₀be ℚ, E or ℍ_ℚ, and let (V,h) be a positive definite K₀-Hermitian space containing a three-system L. Suppose that its projectors span Herm(V,h)⊗_ℚℝ over ℝ. Then

M_1/2(L;ℚ)/M^⊥≃Herm(V,h).

Proof. The projectors and the projector homomorphism are defined over ℚ. Its rational rank equals its real rank, so it is onto. The trace-form identity and positivity after extending to ℝ give kerΨ = M^⊥. □

The projector-span hypothesis is essential. A real three-system placed in a complex space can span the underlying complex vector space while its projectors span only a real symmetric matrix algebra.

Theorem 3.4 (rational factors).The last column ofTable 1gives the Jordan factors over ℚ. Over any field F of characteristic zero, the factors are obtained by tensoring that column with F. Together with Theorem 3.2, this answers [15, Problem 5.1].

Proof. We apply Lemma 3.3 to rational coordinate models. The D_m lines e_i ± e_j have projectors spanning Sym_m(ℚ): their sums and differences give diagonal pair sums and symmetric off-diagonal matrices. Three diagonal pair sums isolate any diagonal entry. Likewise, the three lines e_i − ω^te_j on each support pair in G(3,3,m) give its two-dimensional rational off-diagonal space and its diagonal pair sum. This proves both families for every m ≥ 4.

The standard E₈ coordinates are rational. For E₆, take the subsystem defined by x₆ = x₇ = −x₈. It contains 40 roots of type D₅ and 32 half-integral roots. Its rational span has basis

e₁,…,e₅,e₆ + e₇ − e₈,

with Gram matrix diag(1,1,1,1,1,3). For E₇, use the roots ±2e_i and the vectors with entries ±1 on the complement of a Fano-plane line, and zero elsewhere. These give 63 lines in standard ℚ⁷. The projector ranks for E₆,E₇,E₈ are 21,28,36.

Put θ = ω − ω². The K₆ model consists of the 45 lines e_i − ω^te_j in E⁶ and the 81 distinct lines

$$E\hspace{0.17em}{\theta }^{-1}\left({\omega }^{a_1},\dots ,{\omega }^{a_6}\right), \sum _i{a}_i=0\hspace{0.25em}\left(\mathrm{mod}\hspace{0.25em}3\right).$$

Its 45-line residue perpendicular to (1,…,1) is K₅. Their projector ranks are 36 and 25. To determine the residue form, let ${\left(C_6\right)}_{\mathit{ij}}={\omega }^{a_{\mathit{ij}}}$, where

$$\left(a_{\mathit{ij}}\right)=\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 1& 1& 2& 2\\ 0& 1& 0& 2& 2& 1\\ 0& 1& 2& 0& 1& 2\\ 0& 2& 2& 1& 0& 1\\ 0& 2& 1& 2& 1& 0\end{array}\right).$$

Using 1 + ω + ω² = 0 gives $C_6^{*}{C}_6=6I_6$. The last five columns span (1,…,1)^⊥ and have Gram matrix 6I₅. Its Hermitian form is therefore similar to the standard form. Scalar multiples have the same adjoint involution, so the K₅ factor is Herm₅(E).

Appendix A gives the S₁ realization in standard ${{\mathbb{H}}_{\mathbb{Q}}}^4$. Embed it in ${{\mathbb{H}}_{\mathbb{Q}}}^5$, adjoin e₅ and

$$w=\left({\displaystyle \frac{1-i}{2}}+j,{\displaystyle \frac{1+i}{2}},{\displaystyle \frac{1+i}{2}},-{\displaystyle \frac{1+i}{2}},1\right),$$

and take reflection closure. The resulting U system has 165 lines, projector rank 45, and residue S₁ at e₅. Thus its factor is Herm₅(ℍ_ℚ). Exact coordinate calculations verify these finite angle, closure and rank assertions. The names of the systems follow from the three-system classification used in Theorem 3.1.

Finally, the multiplication and Frobenius matrices are rational, so

$$M_F=M_{\mathbb{Q}}{\otimes }_{\mathbb{Q}}F, M_F^{\perp }=M_{\mathbb{Q}}^{\perp }{\otimes }_{\mathbb{Q}}F.$$

The ideal generated by the polarized Jordan identity also commutes with scalar extension: its generators are multilinear and all multiplication maps are rational. This proves both assertions over F. □

The symmetric-matrix description in [9, Remark 3.5(i)], repeated in the introduction of [15], requires a qualification for E₆. Its factor is Herm₆(F,q₆), which is isomorphic to Sym₆(F) exactly when 3 is a square in F. Indeed, these self-adjoint Jordan algebras are isomorphic exactly when their quadratic forms are similar [23; 29]. In even dimension the determinant square class is a similarity invariant, so similarity to the identity form forces 3 to be a square. The converse follows directly from q₆=⟨1,1,1,1,1,3⟩. In particular, the rational factor is not Sym₆(ℚ).

In Table 1 after scalar extension, E⊗F = F[t]/(t² + 3) carries its canonical involution. If this algebra splits, Herm_r(E⊗F)≃M_r(F)⁺. If ℍ_ℚ⊗F splits, Herm_r(ℍ_ℚ⊗F) is the Jordan algebra fixed by a symplectic involution on M_2r(F).

Two consequences describe the same least ideal from different viewpoints. The first reads positivity and degeneracy from the collinearity spectrum. The second shows that requiring power-associativity already forces the Jordan quotient.

Corollary 3.5 (spectral characterization).Let (G,D) be a finite connected three-transposition group. The Frobenius form of M_1/2(G,D;ℝ) is positive semidefinite with nonzero radical if and only if (G,D) has one of the two families or seven exceptional central types in Table 1.

Proof. Its Gram matrix is I + A/4, where A is the adjacency matrix of the noncommuting graph. The stated condition is equivalent to the least eigenvalue of A being −4. Hall–Shpectorov’s classification [20, Section 7.3.3] gives precisely the two families and seven exceptional central types in Table 1. □

Degeneracy is essential here. The complete diagrams of 3^u:2 have a positive-definite Frobenius form at η = 1/2, but for u ≥ 3 their Matsuo algebras are not Jordan [9].

An algebra is power-associative if the subalgebra generated by each element is associative. Although this is weaker than the Jordan identity in general, it gives the same least ideal in the nine cases.

Corollary 3.6 (power-associative quotients).In each of the nine cases, over every characteristic-zero field, M^⊥is also the least ideal whose quotient is power-associative.

Proof. The quotient by M^⊥ is Jordan, hence power-associative. Conversely, let I be a proper ideal such that M/I is power-associative. Since I ⊆ M^⊥, the generating axes remain nonzero in the quotient. Their multiplication operators remain semisimple, their 1-eigenspaces remain one-dimensional, and their eigenspaces satisfy the same fusion law. Thus M/I is a primitive axial algebra of Jordan type 1/2. Its power-associativity gives the identities x²x² = xx³ and x³x² = xx⁴. Over a characteristic-zero field these polynomial identities hold after every scalar extension. Segev’s Main Theorem and Remark 1.3 therefore imply that M/I is Jordan. Theorem 3.2 gives M^⊥ ⊆ I. □

4. Frame Relations

The equality I_J = M^⊥ identifies the ideal, but does not yet give convenient generators. The projector model suggests comparing orthogonal bases: the sum of their projections depends only on the subspace they span. These relations generate the radical in every case of Table 1 except the G(3,3,m) family. We then examine the frame geometry of S₁.

4.1. Relations for the Classified Systems

Let L be a three-system, and let a_𝓁 denote its Matsuo basis. For an orthogonal set F ⊆ L, write $f_F=\sum _{\mathcal{l}\in F}{a}_{\mathcal{l}}$. A frame relation is a difference f_F − f_F' where the two orthogonal sets span the same subspace. The sets need not be full frames of the ambient space.

Theorem 4.1.Every frame relation lies in the Frobenius radical. For D_m (m ≥ 4), E₆,E₇,E₈,K₅,K₆,S₁,U, these relations span the radical. In the real systems D_m,E₆,E₇,E₈, relations inside D₄ subsystems already suffice. For G(3,3,m) with m ≥ 4 there are no nonzero frame relations; its radical is instead spanned by the star relations

σ_ij + σ_kl − σ_ik − σ_jl  (i,j,k,l distinct),

where σ_ij is the sum of the three axes on e_i − ω^te_j, t = 0,1,2. For A_n and G(3,3,3) the radical is zero.

Proof. The sum of the projectors associated with an orthogonal set is the orthogonal projection onto its span. Thus every frame relation lies in the projector kernel, which is the Frobenius radical. Their span is invariant under the reflection group.

For D_m, the radical description in the least-ideal theorem is the irreducible two-subset module S^(m−2,2). It contains the nonzero relation

x_ij + x_kl − x_ik − x_jl,

where x_ij is the sum of the two axes on e_i + e_j and e_i − e_j. This compares two orthogonal bases of the same coordinate four-space. Its permutation orbit therefore spans the radical and consists of D₄ relations.

For E₆,E₇,E₈, a D₄ subsystem supplies such a nonzero relation. The span of all D₄ relations is invariant and the radical is irreducible under the rank-three reflection action, so that span is the whole radical. This also explains why full ambient frames are unnecessary: E₆ has no orthogonal frame of size six, but the four-dimensional relations suffice.

For K₅ and K₆, the columns of C₆ in Theorem 3.4 give full orthogonal frames: each exponent column sums to zero modulo three, so all six columns represent K₆ lines, and the last five lie in its K₅ residue. In U, the lines on e₁ ± e₂, e₃ ± e₄ and e₅ form a full frame. If F is a full frame, choose a line 𝓁 ∉ F. It meets some member m ∈ F at 60^∘, since F spans the ambient space. Its reflection sends F to a different full frame: r_𝓁(m) is at 60^∘ to m and cannot be another member of F. Hence a nonzero frame relation exists. Its invariant span is the full radical, by the rank-three irreducibility already proved.

For S₁, the nine disjoint frames give eight independent differences. Proposition 4.3 identifies their span with the eight-dimensional radical.

For G(3,3,m), two distinct lines are orthogonal exactly when their index pairs are disjoint. An orthogonal set therefore has disjoint two-coordinate supports. Any line of L in its span must be supported on one of those pairs and must coincide with the selected line there. Consequently two orthogonal sets with the same span are equal, and there are no nonzero frame relations. On the other hand, the radical is the irreducible two-subset module on the sums σ_ij. The displayed nonzero four-index star relation belongs to it, so its permutation orbit spans the radical.

Finally the Gram forms for A_n and G(3,3,3) are nonsingular, giving the final assertion. □

The D₄ example in Figure 2 makes the frame relations explicit: three orthogonal frames give two independent relations, accounting for the entire radical.

, the displayed coefficient columns give the radical basis f₁ − f₂,f₁ − f₃. Each frame has projector sum I₄; the quotient is Sym₄(ℚ).

These relations also characterize functions on the line set that come from self-adjoint operators. A linear functional on the Matsuo algebra descends to the projector quotient exactly when it annihilates the radical.

Corollary 4.2 (frame functions).For D_m (m ≥ 4), E₆,E₇,E₈,K₅,K₆,U,S₁, a real function f on the lines has equal sums on orthogonal sets with the same span if and only if

f(𝓁) = Retr(HP_𝓁)

for a unique self-adjoint operator H in the corresponding Jordan factor.

Proof. The frame conditions say that f, viewed as a linear functional on the Matsuo algebra, vanishes on the frame-relation space. By Theorem 4.1 this is the projector kernel. The trace pairing represents the descended functional by a unique H. The converse follows from equality of projector sums. □

For G(3,3,m) with m ≥ 4, the frame conditions are vacuous, whereas the projector image has codimension m(m − 3)/2 in the space of all functions on the lines.

Full frames have a useful graph-theoretic interpretation. For the systems with nonzero radical, the noncommuting graph has valency κ = 4(N/r − 1) and least eigenvalue −4. Hoffman’s bound limits the size of a coclique, or set of pairwise nonadjacent vertices, to 4N/(κ + 4) = r. A full orthogonal frame attains this bound, and its centred indicator lies in the radical. Each line outside the frame meets exactly four of its members nonorthogonally: pair its projector with the sum of the frame projectors, which is I.

For K₅ and U, take the system’s lines as points and its full orthogonal frames as blocks. These incidence structures are generalized quadrangles of orders (4,2) and (4,8). Each frame has five members, and every point outside it is orthogonal to exactly one member. The rank-three action is transitive on orthogonal pairs, so every such pair lies in a full frame. Distinct frames meet in at most one point: otherwise a point of one outside the other would be orthogonal to two members of the latter. This proves the generalized-quadrangle axioms. Each point has N/5 + 3 orthogonal neighbours, partitioned into sets of four by the frames through it. Thus there are three or nine frames through each point, giving 27 or 297 frames in total.

Table 2 records the full-frame counts and relation ranks. Exact enumeration verifies these values; the E₈ counts also occur in the Kochen–Specker construction of Waegell–Aravind [36].

4.2. The Nine Quaternionic Frames

The S₁ radical is reducible under its reflection group. The case-9 proof in Theorem 3.2 uses a larger group of automorphisms that acts irreducibly. The nine orthogonal frames make both actions explicit.

Proposition 4.3 (the frame radical of S₁).The 36 lines of the quaternionic system S₁ have nine disjoint full frames F₁,…,F₉, and two distinct lines are orthogonal exactly when they belong to the same frame. For M = M_1/2(S₁),

$$M^{\perp }=\mathrm{span}\left\{f_{F_i}-f_{F_9}:1\le i\le 8\right\}.$$

In particular, dimM^⊥ = 8, and the projector homomorphism identifies M/M^⊥ with Herm₄(ℍ) over ℝ.

Proof. The frame description and the orthogonality assertion are given by Griffeth–Guevara [16]. The nonorthogonality adjacency matrix is therefore

A = J₃₆ − diag(J₄,…,J₄).

It acts by 32 on constants, by −4 on block-constant vectors whose nine block values sum to zero, and by 0 on vectors having sum zero within each block. These subspaces have dimensions 1,8,27, respectively. The Frobenius Gram matrix I + A/4 consequently has spectrum 9⁽¹⁾,0⁽⁸⁾,1⁽²⁷⁾. Its kernel is precisely the displayed frame-sum space. The projector image has dimension 28, while dim_ℝHerm₄(ℍ) = 28; hence the homomorphism is onto. □

The nine frames are mutually unbiased: the trace product of projectors from distinct frames is 1/4. They form a complete set, since their three-dimensional traceless projector spaces are mutually orthogonal in the 27-dimensional traceless part of Herm₄(ℍ). Nine quaternionic mutually unbiased bases in dimension four already occur in [22, Example 21]; see also [5].

To describe the reflection action on the radical, label the frames by the points of an affine plane. The 192 Fischer lines involve twelve triples of frames, with sixteen Fischer lines per triple. Each triple supports a D₄ subsystem, and these twelve triples form the lines of AG(2,3) [16]. Its four parallel classes will index the radical summands.

Proposition 4.4 (the four radical summands).Index the nine frames by$p\in {\mathbb{F}}_3^2$and put$b_p=\sum _{\mathcal{l}\in F_p}{a}_{\mathcal{l}}$. For a direction d, with parallel affine lines L₁,L₂,L₃, define

$$R_d=\{{\displaystyle \sum _{i=1}^3}{c}_i\sum _{p\in L_i}{b}_p:\hspace{0.25em}{\displaystyle \sum _{i=1}^3}{c}_i=0\}.$$

Then M^⊥ = ⨁_dR_d. The four summands have dimension two and are irreducible over ℚ and ℝ under the reflection group. Its action on the nine frames is 3²:2.

Proof. The frame sums identify the radical with the augmentation module on ${\mathbb{F}}_3^2$. A reflection belonging to F_p acts on frame labels by q ↦ 2p − q, so the induced group consists of translations and inversion. It preserves each parallel class. A translation outside d cyclically permutes its three affine lines, acting on R_d with minimal polynomial X² + X + 1, irreducible over ℚ and ℝ. For the auxiliary inner product making the b_p orthonormal, distinct R_d are orthogonal because affine lines of distinct directions meet once. Their dimensions sum to eight. □

A single difference b_p − b_q generates precisely the three summands whose directions differ from ⟨p − q⟩ (Figure 3). Two differences in distinct directions generate the radical. A general radical vector can have nonzero components in all four summands. The reflection action has rank five on frames and rank six on the 36 axes.

The full automorphism action is larger. Realize S₁ as the residue perpendicular to a line 𝓁 in U, as in the proof of Theorem 3.4. The stabilizer of 𝓁 in the reflection group of U induces automorphisms of the residue. The induced frame group is ASL(2,3) of order 216: exact Schreier enumeration through all 165 lines gives affine maps on the nine frames with determinant-one linear parts, and their closure has order 216. Its action is two-transitive. Its permutation action on the four parallel classes is transitive, with kernel 3²​:​2, precisely the frame action of the S₁ reflections. Thus the frame augmentation module M^⊥ is absolutely irreducible under this larger group, as used in Theorem 3.2.

4.3. Kantor’s Quaternionic Line Sets

The nine mutually unbiased frames suggest comparing S₁ with Kantor’s quaternionic line sets. This comparison also tests whether the angle and orthogonality data determine a three-system. The enumeration below finds two configurations with those data, only one of which is reflection-closed.

Kantor’s construction [26, Sections 4–6 and 8.3] associates quaternionic line sets to orthogonal spreads. In dimension four, the extraspecial group P = Q₈∘D₈∘D₈ acts on ℍ⁴: Q₈ acts by left scalar multiplication, while the two dihedral factors act by the real Pauli matrices on ℝ²⊗ℝ². The quotient P/Z(P) is a six-dimensional quadratic space of minus type over 𝔽₂. Its 27 nonzero singular vectors lie in 45 totally singular two-spaces. An orthogonal spread partitions these vectors into nine such spaces. The inverse image of each two-space in P is an elementary abelian group of order eight; its four joint eigenspaces on which the central involution acts as −1 are quaternionic lines forming an orthogonal frame. The nine frames are mutually unbiased.

Proposition 4.5 (exact enumeration).The quadratic space has 200 orthogonal spreads, in two$O_6^{-}\left(2\right)$-orbits of sizes 40 and 160. The orbit of size 40 gives S₁. The other orbit gives an inequivalent configuration with the same orthogonality graph 9K₄ and the same 576 unordered pairs at 60^∘. For 432 of these pairs, the third line lies outside the configuration. Its reflection closure has 72 lines.

Proof. Exact enumeration gives all partitions of the 27 singular vectors by nine of the 45 singular two-spaces. The 36 symmetries indexed by nonsingular vectors generate an orthogonal group of order 51 840; their action on the 200 partitions has the stated orbit sizes. For every spread, separate integer-quaternion calculations give 36 distinct lines, 54 unordered orthogonal pairs and 576 pairs at 60^∘. All third lines remain in the set for the 40 spreads; exactly 432 fail this test for each of the other 160.

The unitary matrix 2^−1/2(H₂⊕H₂), where $H_2=\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$, carries the rational S₁ model of Appendix A to one of the closed configurations. Kantor’s equivalence theorem [26] identifies equivalence of the line sets with equivalence of their spreads. Finally, closing one representative of the second orbit under all reflections until no new line appears gives 72 lines. Unitary equivalence gives the same closure size throughout that orbit. □

The enumeration also describes closure at the level of frames (Figure 4). In each of the 160 nonclosed spread configurations, the nine frames split into three triples. The twelve lines belonging to each triple form a reflection-closed set. For frames in different triples, all sixteen cross-frame line pairs have their third line outside the configuration. Thus the 27 unordered frame pairs between triples account for all 27 ⋅ 16 = 432 failures. In each of the 40 closed configurations, every pair of distinct frames closes. These assertions were checked exactly for all 200 spreads.

5. The Special Bound and Projective Designs

The projector realization places the systems of Table 1 among line sets subject to classical size bounds. We recall the special bound of Delsarte–Goethals–Seidel and Hoggar [10, 21] and show how its equality cases meet the Jordan-factor classification. This section is a consequence of the geometric description and is not needed for the least-ideal theorem.

Let L be an indecomposable three-system of N lines spanning K^r, with r ≥ 2, and put Δ=dim_ℝK. A projective s-design is a finite line set whose projector tensor moments through degree s equal those of the invariant probability measure on projective space. All projector tensor products in this section are over ℝ, including when K = ℍ. The one-line case is a design in every degree and is excluded from the criterion below. The moment proof makes the connection with the Matsuo Gram matrix explicit.

Proposition 5.1.If Δr < 4Δ+6, then

$$N\le {\displaystyle \frac{3r\left(\Delta r+2\right)}{4\Delta +6-\Delta r}}.$$

Equality holds exactly for a projective 2-design. Among equality cases, the projective 3-design condition is Δr = 4(Δ+1).

Proof. Write P_𝓁 for the unit rank-one projections. The operator $\sum P_{\mathcal{l}}$ commutes with the irreducible reflection group, so it is (N/r)I. Taking the trace pairing with one projector shows that each line has κ = 4(N/r − 1) nonorthogonal partners. Since ⟨P_𝓁,P_m⟩ takes the values 1,0,1/4,

$$E_s:={\displaystyle \frac{1}{N^2}}\sum _{\mathcal{l},m}⟨P_{\mathcal{l}},P_m{⟩}^s={\displaystyle \frac{1+4^{1-s}\left(N/r-1\right)}{N}}.$$

To compare these finite moments with projective space, let μ be its invariant probability measure and put $T_s=\int P^{\otimes s}\hspace{0.17em}\mathit{d\mu }\left(P\right)$. The squared first-coordinate norm of a random unit vector has beta distribution with parameters Δ/2,Δ(r − 1)/2. Hence, with (x)_s = x(x + 1) … (x + s − 1),

$$c_s={\displaystyle \frac{{\left(\Delta /2\right)}_s}{{\left(\Delta r/2\right)}_s}}, {\parallel {\displaystyle \frac{1}{N}}\sum _{\mathcal{l}}{P}_{\mathcal{l}}^{\otimes s}-T_s\parallel }^2=E_s-c_s.$$

Thus E_s ≥ c_s, with equality exactly when the configuration is a projective s-design. At s = 2 this is

$${\displaystyle \frac{N+3r}{4\mathit{rN}}}\ge {\displaystyle \frac{\Delta +2}{r\left(\Delta r+2\right)}},$$

which gives the bound. Substituting its equality value of N at s = 3 gives

$$E_3-c_3={\displaystyle \frac{\Delta \left(r-1\right)\left(4-\Delta \left(r-4\right)\right)}{4r\left(\Delta r+2\right)\left(\Delta r+4\right)}}.$$

For r ≥ 2, this vanishes exactly when Δr = 4(Δ+1). □

In the minimal realizations of Theorem 3.1, the equality cases are the seven exceptional systems in Table 1 and Table A2, D₄,G(3,3,3). Among them, E₈,K₆,U are the classical tight projective 3-designs [2, 22]; Hoggar’s Examples 7 and 9 are K₆ and U, respectively. Equivalently, the antipodal E₈ roots form a spherical 7-design. The condition Δr = 4(Δ+1) must be used together with equality in the bound: D₈ and G(3,3,6) satisfy that numerical condition but fall below the bound.

6. Incidence Coordinates for the Symplectic Family

The E₇ Fischer space is also the transvection space of Sp₆(2). This allows its radical and quotient to be described by incidence with quadratic forms, and suggests extending the construction to Sp_2n(2). The spectrum, degeneration parameters and dimensions are known [20, row PR4; 33, Sections 5 and 9]. Here we construct explicit maps onto the quotient modules, calculate their forms and give incidence generators for the radicals. The resulting quotients need not be Jordan.

Let V be a 2n-dimensional vector space over 𝔽₂ with nondegenerate alternating form b, where n ≥ 3. Put

t = 2ⁿ⁻¹,  X = V∖{0},  v = ∣X∣ = 4t² − 1.

The transvection associated with x ∈ X is z ↦ z + b(z,x)x. Distinct transvections do not commute precisely when b(x,y) = 1; the corresponding Fischer line is {x,y,x + y}. Let A be this adjacency matrix. Write J for an all-ones matrix, with dimensions determined by its position. We use the product (1) and the normalized Gram matrix G_η = I + (η/2)A.

We use quadratic forms with polarization b to index quotient coordinates. Their evaluation vectors will also generate the radicals. For ε ∈ {+1,−1}, let Ω_ε be the forms with character sum $\sum _{z\in V}{\left(-1\right)}^{q\left(z\right)}=\epsilon 2t$; thus + and − denote plus and minus type. Set

m_ε = ∣Ω_ε∣ = t(2t + ε),  k_ε = t(2t − ε).

Here m_ε counts forms of type ε, and k_ε counts the nonzero vectors at which any such form takes the value one. Let B_ε have rows indexed by x ∈ X, columns indexed by q ∈ Ω_ε, and entries ${\left(B_{\epsilon }\right)}_{x,q}=q\left(x\right)$, regarded as rational numbers 0 or 1. The matrix products below determine its rank and the adjacency eigenspace spanned by its centred columns. We write 1 for an all-ones vector of the indicated size; the augmentation subspace consists of vectors whose coordinates sum to zero.

Theorem 6.1 (symplectic incidence identities).The matrices B_ε satisfy

$$B_{\epsilon }1=t^{2}1, B_{\epsilon }^{T}1=k_{\epsilon }1,$$

$$B_{\epsilon }^T{B}_{\epsilon }=t^{2}I+t\left(t-\epsilon \right)J,$$

$$B_{\epsilon }{B}_{\epsilon }^T={\displaystyle \frac{t^2}{2}}\left(I+J\right)-{\displaystyle \frac{\mathit{\epsilon t}}{2}}A.$$

(3)

In particular, B_ε has column rank m_ε. The centered matrix

$$C_{\epsilon }=B_{\epsilon }-{\displaystyle \frac{t}{2t+\epsilon }}J$$

has rank m_ε − 1 and its column space is the −εt eigenspace of A.

Proof. A nonzero linear functional on V takes the value 1 on 2t² vectors. Two distinct nonzero linear functionals are independent, so their simultaneous value 1 occurs on t² vectors. Hence

A1 = 2t²1,  A² = t²(I + J).

Its three eigenvalues and multiplicities are therefore

$$\mathrm{spec}\left(A\right)=\left\{{\left(2t^2\right)}^1,\hspace{0.25em}{t}^{\hspace{0.17em}{m}_{-}-1},\hspace{0.25em}{\left(-t\right)}^{\hspace{0.17em}{m}_{+}-1}\right\}.$$

For the incidence counts, choose a plus-type form q₀. Every quadratic form with polarization b has the unique expression q_a = q₀ + b(a,⋅), and its type is $\epsilon ={\left(-1\right)}^{q_0\left(a\right)}$. Completing the square gives

$$\sum _{a\in V}{\left(-1\right)}^{q_0\left(a\right)+b\left(a,x\right)}=2t{\left(-1\right)}^{q_0\left(x\right)}.$$

Consequently, for x ≠ 0, $\sum _{q\in {\Omega }_{\epsilon }}{\left(-1\right)}^{q\left(x\right)}=\mathit{\epsilon t}$. This gives t² ones in each row of B_ε; each column has k_ε ones by the definition of type. For distinct x,y ∈ X, the identity

(−1)^q(x)+q(y) = (−1)^b(x,y)(−1)^q(x+y)

gives t²/2 common ones when b(x,y) = 0, and (t² − εt)/2 when b(x,y) = 1. These give (3), the entries of $B_{\epsilon }{B}_{\epsilon }^T$. For distinct forms q,r of the same type, q + r is a nonzero linear functional. Expanding the indicators of q(x) = r(x) = 1 gives t² − εt common ones, proving the other matrix identity. Its eigenvalues are t² on the augmentation space and t³(2t − ε) on constants, both nonzero. Finally, centering removes the constant component, and the second identity acts by t² on the −εt eigenspace and by zero on the other nonconstant eigenspace. Thus imC_ε is exactly the claimed eigenspace. □

Corollary 6.2 (both nontrivial degenerations).The Gram determinant is

$$\det G_{\eta }=\left(1+\eta t^2\right){\left(1+{\displaystyle \frac{\mathit{\eta t}}{2}}\right)}^{m_{-}-1}{\left(1-{\displaystyle \frac{\mathit{\eta t}}{2}}\right)}^{m_{+}-1}.$$

At η = 2ε/t,

$$\mathrm{rad}{G}_{2\epsilon /t}=\mathrm{im}{C}_{\epsilon }=\ker B_{-\epsilon }^T.$$

Thus the radical is generated by centered incidence vectors of the form type opposite to the form type indexing the quotient. The map $B_{-\epsilon }^T$ induces an Sp_2n(2)-equivariant isomorphism

$$M_{2\epsilon /t}\left({\mathrm{Sp}}_{2n}\left(2\right)\right)/\mathrm{rad}{G}_{2\epsilon /t}\hspace{0.25em}\cong \hspace{0.25em}{\mathbb{Q}}^{{\Omega }_{-\epsilon }}.$$

In these coordinates the induced bilinear form is

$$⟨u,w⟩={\displaystyle \frac{2}{t^2}}{u}^{T}w-{\displaystyle \frac{1}{t^4}}\left(1^{T}u\right)\left(1^{T}w\right).$$

Proof. The determinant and radical follow from Theorem 6.1. The incidence matrices are equivariant because their entries are evaluations of quadratic forms. With B = B_−ε, the matrix identities give

$$G_{2\epsilon /t}={\displaystyle \frac{2}{t^2}}B{B}^T-J.$$

For u = B^Tx, the row sum identity gives 1^Tu = t²1^Tx, yielding the stated form. The radical is an ideal because the Matsuo form associates, so the vector-space quotient also has its inherited algebra product. The displayed identification specifies the group module and the form; it does not identify that product with coordinatewise multiplication. □

On the augmentation space the quotient form has eigenvalue 2/t². On constants it has eigenvalue ε/t³. Thus the positive degeneration has a positive definite quotient form, proved directly by incidence counting. The negative degeneration has one negative direction. The third degeneration is η = −1/t², with radical ℚ1; it is not one of these two incidence quotients.

For n = 3, the radical at η = 1/2 has dimension 35 and is spanned by the 36 centred evaluation vectors B_+,q − (4/9)1, q ∈ Ω₊. Its quotient has the 28 minus-type forms as coordinates. At η = −1/2, the radical has dimension 27 and is spanned by B_−,q − (4/7)1, q ∈ Ω₋. The quotient then has dimension 36. Section 7 realizes the positive degeneration in E₇ roots.

For n = 4, the parameters are η = ±1/4. The positive quotient has dimension 120 and radical dimension 135; the negative quotient has dimension 136 and radical dimension 119. Matsuo’s vertex operator algebra classification already contains the positive case, with quotient dimension 120 and central charge 15/2 [32, Table 3]. Its VOA construction is due to Kitazume and Miyamoto [28; 32, Section 8]. The present maps supply incidence coordinates for this quotient.

Remark 6.3.The positive-degeneration quotient is non-Jordan for n ≥ 4, and the negative-degeneration quotient is non-Jordan for every n ≥ 3. This follows from [15, Lemma 5.1]: at η ≠ 1/2, a nonzero Jordan quotient of a connected Matsuo algebra has dimension one.

The counting identities remain valid for n = 2, when the two nonconstant degenerations occur at η = ±1. The restriction n ≥ 3 keeps the discussion within the usual Matsuo parameter range η ≠ 0,1.

There is a second source of radical generators, analogous to the full-frame relations of Section 4. A Lagrangian is an n-dimensional totally isotropic subspace of V; its nonzero vectors index pairwise commuting transvections.

Proposition 6.4 (Lagrangian relations).For every n ≥ 3, the radical of $M_{2^{2-n}}\left({\mathrm{Sp}}_{2n}\left(2\right)\right)$ is spanned by

$$\sum _{x\in L\setminus \left\{0\right\}}{a}_x-\sum _{x\in L\mathrm{\text{'}}\setminus \left\{0\right\}}{a}_x,$$

where L,L' range over the Lagrangian subspaces of ${\mathbb{F}}_2^{2n}$.

Proof. Put ${\Lambda }_m={\displaystyle \prod _{i=1}^m}\left(2^i+1\right)$ and Λ₀ = 1. Let T be the nonzero-point by Lagrangian incidence matrix. Counting Lagrangians through a point and through a perpendicular pair gives

TT^T = Λ_n−2(tI + J − A),  t = 2ⁿ⁻¹.

(4)

On the two nonconstant adjacency eigenspaces, (4) acts by 0 at eigenvalue t and by 2tΛ_n−2 at eigenvalue −t. The centred columns $T_L-{\left(2^n+1\right)}^{-1}1$ therefore span the −t eigenspace. Their sum is zero, so their span equals that of the column differences. This eigenspace is the kernel of I + A/t, the Frobenius Gram matrix at η = 2/t. □

The Lagrangian cocliques have 2ⁿ − 1 points and attain Hoffman’s bound. Exact enumeration gives all 135 and 2295 Lagrangians for n = 3,4 and relation ranks 35 and 135, respectively.

7. The E₇ Worked Example

We now specialize Section 6 to n = 3, where the positive degeneration is the Jordan quotient attached to E₇. The 63 root lines index the Matsuo axes, while 28 pairs of minuscule weights index the incidence coordinates. Identifying these pairs with the minus-type quadratic forms explains the quotient map and its form. Appendix C gives the multiplication in these coordinates and compares the classical bitangent and equiangular models [31, 35, 11, 3].

7.1. Roots and Minuscule Weight Pairs

Let e₁,…,e₈ be the standard orthonormal basis of ℚ⁸, put $s=\sum _i{e}_i,$ and let H₇ = s^⊥. A root system of type E₇ in H₇ is

$$\Phi =\{e_i-e_j:i\ne j\}\hspace{0.33em}\cup \hspace{0.33em}\{r_S={\displaystyle \frac{1}{2}}\left(\sum _{i\in S}{e}_i-\sum _{i\notin S}{e}_i\right):|S|=4\}.$$

All roots have squared length 2. The 63 root lines are indexed by 28 differences with i < j and 35 unordered partitions {S,S^c}. We denote this set by $ℛ$.

To obtain the 28 quotient coordinates, use the 56 minuscule weights ±λ_p, where

$${\lambda }_p=\sum _{i\in p}{e}_i-{\displaystyle \frac{1}{4}}s, p\in \mathcal{P}=\left({\displaystyle \genfrac{}{}{0pt}{}{\{1,\dots ,8\}}{2}}\right).$$

The set $𝒫$ indexes their 28 sign pairs. Coordinate permutations preserve these weights. Reflection in r_S fixes λ_p when p crosses the partition S∣S^c; when p ⊆ S, it sends λ_p to −λ_q, where q is the two-element complement of p in S, and similarly for p ⊆ S^c. Thus the Weyl action on weight pairs is explicit.

Orthogonality between root lines and weight pairs defines the 63 × 28 incidence matrix

$$B_{\mathcal{l},p}=\{\begin{array}{ll}1,& \alpha \cdot {\lambda }_p=0\mathrm{for}\alpha \in \mathcal{l},\\ 0,& \mathrm{otherwise}.\end{array}$$

For 𝓁 = ℚ(e_i − e_j), incidence means that p contains either both or neither of i,j. For 𝓁 = ℚr_S, incidence means that p crosses S∣S^c. These rules define B over the integers.

Let 𝒜 be the adjacency matrix on distinct root lines, with adjacency meaning nonorthogonality. Write J_a,b for the all-ones a × b matrix and J_a = J_a,a.

To identify this root-weight incidence with the construction of Section 6, pass to a binary space. Let V₆ be the six-dimensional space of even subsets of {1,…,8} modulo complementation, with symmetric difference as addition. The forms

b(S,T) = |S ∩ T| (mod 2),  q₀(S) = |S|/2 (mod 2)

are well-defined, and b is the nondegenerate polarization of q₀. Sending a difference root line to {i,j} and a partition root line to S identifies $ℛ$ with V₆∖{0}. Nonorthogonality is exactly b(x,y) = 1. Thus 𝒜 is the symplectic graph with parameters (63,32,16,16): a nonzero linear functional takes value 1 on 32 vectors, and two distinct such functionals are independent and simultaneously take value 1 on 16 vectors. Consequently

𝒜² = 16I + 16J,  spec𝒜 = {32¹,4²⁷,(−4)³⁵}.

The multiplicities follow from total dimension and trace zero. This is the standard symplectic graph spectrum, also tabulated by Matsuo [32]. The classical identification W(E₇)/{±1}≅Sp(6,2) is compatible with this model: root reflections act as x ↦ x + b(x,y)y, and these transvections generate the symplectic group [17, 1].

The 35 nonzero zeros of q₀ form the hyperbolic quadric Q⁺(5,2), and its 28 nonsingular vectors are the two-subsets p. For each such p, put q_p(x) = q₀(x) + b(p,x). These are the 28 quadratic forms of minus type with polarization b; each has 27 nonzero zeros and 36 ones. In our coordinates the incidence rule is simply B_x,p = q_p(x), regarded as 0 or 1.

Theorem 7.1.The incidence matrix satisfies

B1₂₈ = 161₆₃,  B^𝖳1₆₃ = 361₂₈,

B^𝖳B = 16I₂₈ + 20J₂₈,  BB^𝖳 = 8I₆₃ + 2𝒜 + 8J₆₃.

(5)

Proof. The identities (5) are the minus-type identities of Theorem 6.1 at n = 3. In the present coordinates, a row has 16 ones, a column has 36 ones, distinct columns meet in 20 rows, and distinct rows meet in 8 or 10 columns according as the roots are orthogonal or nonorthogonal. □

Equivalently, B is the block-by-point incidence matrix of a 2-(28,16,20) design with 63 blocks and replication number 36. Its complement is the classical 2-(28,12,11) Steiner design discussed in Appendix C.3.

7.2. The Frobenius Quotient and Its Form

The incidence matrix now gives the quotient map. Write γ_M,Δ_M for Matsuo’s parameters: his Gram matrix is (γ_M/2)(I + (Δ_M/4)𝒜), so our parameter is η = Δ_M/2 after normalizing the diagonal to one [32, Theorem 3.2.5, Table 5 and Note 4.2.5]. At η = 1/2, its radical is the 35-dimensional −4 eigenspace of 𝒜. The following theorem expresses the resulting quotient and form in the 28 weight-pair coordinates.

Theorem 7.2 (incidence coordinates).At η = 1/2, the map

Φ_B:ℚ^$ℛ$ → ℚ^$𝒫$,  x ↦ B^𝖳x

is surjective, is Weyl-equivariant, and has kernel radG_1/2. Hence it induces a canonical isomorphism of Weyl modules

M_1/2(E₇)/radG_1/2 ≅ ℚ^$𝒫$.

The quotient form in these coordinates is

$$⟨u,v{⟩}_Q={\displaystyle \frac{1}{8}}{u}^{\mathsf{T}}v-{\displaystyle \frac{1}{256}}\left(1^{\mathsf{T}}u\right)\left(1^{\mathsf{T}}v\right).$$

(6)

Proof. The eigenvalues of B^TB = 16I + 20J are 576 on constants and 16 on their orthogonal complement, so B has rank 28. Orthogonality of roots and weights is Weyl-invariant, proving equivariance. Theorem 7.1 gives

$$G_{1/2}={\displaystyle \frac{1}{8}}B{B}^{\mathsf{T}}-J_{63}.$$

If B^Tx = 0, then 161^Tx = 0, whence G_1/2x = 0. Conversely, the row sums in Theorem 7.1 imply 𝒜1 = 321 and G_1/21 = 91. Thus G_1/2x = 0 implies 1^Tx = 0 and then BB^Tx = 0. Multiplying by B^T and using invertibility of B^TB gives B^Tx = 0. Finally, 1^TB^Tx = 161^Tx gives (6). □

The quotient form has eigenvalue 1/64 on constants and 1/8 on the augmentation subspace, so it is positive definite. The adjacency spectrum also gives

detG_η = (1 + 16η)(1 + 2η)²⁷(1 − 2η)³⁵.

The three degeneration parameters are −1/16,−1/2,1/2, with radical dimensions 1,27,35, respectively [32, 20].

The Weyl action on the 28 weight pairs is two-transitive. The stabilizer of p = {1,2} in the coordinate-permutation subgroup 𝔖₈ has two orbits on the other pairs: those meeting p once and those disjoint from it. Reflection in r_{1,3,4,5} fixes p and sends {1,3} to {4,5}, joining these orbits. Hence the augmentation subspace of ℚ^$𝒫$ is absolutely irreducible.

Write V₂₇ and V₃₅ for the eigenspaces of 𝒜 in ℚ^$ℛ$ with eigenvalues 4 and −4. The map B^T identifies V₂₇ with the augmentation subspace of ℚ^P and kills V₃₅. For the other degeneration, put B⁺=B₊, the incidence matrix on plus-type quadratic forms from Section 6. Identifying P with Ω₋ as in Section 7.1 gives the two quotient maps

$$\begin{array}{rr}{\mathbb{Q}}^{\mathcal{R}}\stackrel{B^{\mathsf{T}}}{\to }{\mathbb{Q}}^{{\Omega }_{-}}, & \ker B^{\mathsf{T}}=V_{35},\\ {\mathbb{Q}}^{\mathcal{R}}\stackrel{{\left(B^{+}\right)}^{\mathsf{T}}}{\to }{\mathbb{Q}}^{{\Omega }_{+}}, & \ker {\left(B^{+}\right)}^{\mathsf{T}}=V_{27}.\end{array}$$

The 28-dimensional quotient is Sym₇(ℚ); the 36-dimensional quotient is not Jordan. Appendix C.1 gives the transported multiplication in the 28-dimensional coordinates.

8. Finite Subspaces and Further Questions

For an infinite Fischer space, every algebra element still has finite support. This suggests passing the finite quotient results to a directed union, but the radical of a finite subalgebra need not in general agree with its intersection with the full radical. Positivity from Theorem 3.1 supplies the required compatibility. For a finite Fischer subspace S, write M_S = M_1/2(S;ℝ) and $R_S=M_S^{\perp }$, with the radical taken inside M_S.

Proposition 8.1.Let (G,D) be a possibly infinite three-transposition group, which is locally finite [8], and let M = M_1/2(G,D;ℝ) = ⨁_d∈Dℝa_d be its finite-support Matsuo algebra. Put R = M^⊥. Then M/R is Jordan if and only if M_S/R_S is Jordan for every finite Fischer subspace S ⊆ D. When these conditions hold, the Frobenius form on M is positive semidefinite and

$$R\cap M_S=R_S, R=\bigcup _S{R}_S=I_J\left(M\right).$$

Here the union is directed over the finite Fischer subspaces of D.

Proof. If M/R is Jordan, then M_S/(M_S ∩ R) embeds in it. Since M_S ∩ R ⊆ R_S, the quotient M_S/R_S is also Jordan.

Assume now that every finite quotient is Jordan. Theorem 3.1 makes the Frobenius form positive semidefinite on each non-singleton connected component; the singleton case is immediate. Hence the form on M is positive semidefinite, since every vector has finite support. For a positive semidefinite bilinear form, zero squared length is equivalent to membership in the radical. This proves R ∩ M_S = R_S and the union formula. Every Jordan defect belongs to some M_S and lies in R_S, so M/R is Jordan. On a component with nonzero radical, the finite classification [15, Proposition 5.2] and Theorem 3.2 give R_S = I_J(M_S); on a component with zero radical both sides are zero. Thus R_S ⊆ I_J(M) for every S, proving R = I_J(M). □

Proposition 8.1 reduces the infinite problem to the finite classification.The remaining task is to choose coordinate models that agree under inclusions of finite subspaces. The candidates are the infinite versions of the three unbounded families.

For an infinite set I, let 𝕂^(I) denote the finitely supported coordinate vectors, with their standard inner product, and put $V_A\left(I\right)=\{v\in {\mathbb{R}}^{\left(I\right)}:\sum _i{v}_i=0\}$. Define A(I), D(I) and G(3,3,I) by the lines

ℝ(e_i − e_j),  ℝ(e_i ± e_j),  ℂ(e_i − ω^ae_j) (a = 0,1,2),

respectively, with distinct i,j ∈ I and ω³ = 1 ≠ ω.

Proposition 8.2.An infinite connected locally finite Fischer space has a Jordan quotient by its Frobenius radical if and only if it is isomorphic to A(I), D(I) or G(3,3,I) for some infinite set I. Its real Jordan quotient is the algebra of finite-rank self-adjoint endomorphisms of V_A(I), ℝ^(I) or ℂ^(I), respectively. No completion of these inner-product spaces is taken. For countable I, these are the usual systems A_∞, D_∞ and G(3,3,∞).

Proof. We first describe the connected subspaces of a finite coordinate family. Write its lines as e_i − ζ^ae_j, with gains a ∈ ℤ/qℤ, where q = 1,2,3 and ζ = 1,−1,ω, respectively. A connected Fischer subspace has a connected graph of coordinate supports. Multiplying coordinate vectors by powers of ζ makes the gains on a spanning tree zero. Closure under the third-point operation then gives every zero-gain line on those coordinates. If a nonzero gain occurs, the zero-gain reflections permute the coordinates and propagate that gain to every pair; reversing a pair also gives its negative. For q = 2,3 this gives the entire coordinate family. Thus a connected subspace is a zero-gain A system or a full family of the ambient type.

Every finite set of points lies in a finite connected Fischer subspace: add finitely many paths and use local finiteness. By Proposition 8.1 and the finite three-system classification recalled after Theorem 3.1, a connected subspace with more than 165 points is of type A, D or G(3,3). The preceding coordinate description separates the types. The space D₄ has twelve points, whereas connected subspaces of an A or G(3,3) family have s(s − 1)/2 or 3s(s − 1)/2 points; neither number can be twelve. Similarly, G(3,3,3) has nine points and cannot occur in an A or D family, whose possible connected sizes are s(s − 1)/2 and s(s − 1). If either space occurs, the finite connected subspaces containing it and having more than 165 points form a cofinal family of its type. If neither occurs, all sufficiently large connected subspaces have type A.

In this cofinal family, each inclusion is a coordinate inclusion after relabelling. Indeed the smaller space is a connected subspace of the larger one, and the coordinate description applies; the D₄ or G(3,3,3) subspace excludes the zero-gain alternative in the corresponding cases. The coordinate labels themselves can be made intrinsic. In a D family on at least five coordinates, points with the same open collinearity neighbourhood are precisely the two lines on one coordinate pair. In a G(3,3) family, the analogous closed-neighbourhood classes are the three lines on one pair. Collapsing these classes gives the line graph of a complete graph; for type A the collinearity graph already has this form. Its maximal cliques of size at least four recover the coordinate stars. These stars are preserved under the inclusions, and their directed union is an index set I.

It remains to choose the gains consistently. Each finite subspace has a nonempty finite set of coordinate labellings on its recovered coordinate set. Restriction along an inclusion gives such a labelling on the smaller subspace. Every finite collection of these compatibility requirements can be satisfied by choosing a labelling on a common upper subspace and restricting it. Compactness of a product of finite discrete sets therefore supplies compatible labellings on the whole directed family. They identify the original Fischer space with the stated coordinate system. This argument does not require countability.

Conversely, each of the three coordinate systems is a directed union of its finite models. Proposition 8.1 makes its quotient Jordan. The projector map has kernel equal to the radical by the positive trace-form identity. Its image is the stated algebra: the projectors span each finite coordinate block, and a finite-rank self-adjoint operator is supported on a finite coordinate block containing its range. □

A proof of positivity using only the Matsuo multiplication and finite axis action, without the structure theory of simple Jordan algebras, remains open.

For n ≥ 4, the symplectic incidence map gives an explicit multiplication model by transport through the quotient. An intrinsic description of these non-Jordan algebras would be useful. At n = 4, the positive quotient already has a VOA realization [32, 28]; the next question is to compare the incidence and VOA coordinates.

Appendix C.1 Comparison with the Jordan Realization

Let $𝒥$ be the Jordan algebra of self-adjoint endomorphisms of H₇, with product X∘Y = (XY + YX)/2. For a root α, set

$$P_{\alpha }={\displaystyle \frac{1}{2}}\alpha {\alpha }^{\mathsf{T}}{|}_{H_7}.$$

The map Ψ(a_ℚα) = P_α is the root-projection construction of [9, Lemmas 3.2–3.3]. Its target is the self-adjoint algebra for the rational form on H₇, identified with Sym₇(ℚ) in Theorem 3.4.

Proposition C.1 (De Medts–Rehren, specialized to E₇).The map Ψ:M_1/2(E₇) → $𝒥$ is a surjective algebra homomorphism with kernel radG_1/2.

Proof. Each P_α is an idempotent, and orthogonal roots give zero product. If α ⋅ β = ϵ ∈ {1,−1}, the third line is represented by γ = α − ϵβ, and direct expansion gives

$$P_{\alpha }\circ P_{\beta }={\displaystyle \frac{1}{4}}(P_{\alpha }+P_{\beta }-P_{\gamma }).$$

Thus Ψ preserves the Matsuo product at η = 1/2. The 28 projections corresponding to e_i − e_j are linearly independent. Indeed, a linear combination of their full 8 × 8 matrices vanishing on H₇ also vanishes on s, so is the zero matrix; its (i,j) off-diagonal entry then forces the corresponding coefficient to vanish. Since dim$𝒥$ = 7 ⋅ 8/2 = 28, they span $𝒥$.

Finally,

$$\mathrm{tr}\left(P_{\alpha }{P}_{\beta }\right)={\displaystyle \frac{{\left(\alpha \cdot \beta \right)}^2}{4}}$$

is precisely the Matsuo Gram entry. The trace form on the self-adjoint operators is nondegenerate, and surjectivity of Ψ identifies its kernel with the radical of that Gram matrix. □

To transport multiplication to incidence coordinates, we need to lift a coordinate vector back to the Matsuo algebra. A Weyl-equivariant right inverse of B^T is obtained by setting

$$C=B\left({\displaystyle \frac{1}{16}}{I}_{28}-{\displaystyle \frac{5}{2304}}{J}_{28}\right).$$

The expression in parentheses is ${\left(B^{\mathsf{T}}B\right)}^{-1},$ so B^TC = I₂₈. Consequently u ↦ Ψ(Cu) gives the isomorphism from incidence coordinates to $𝒥$. Equivalently, their transported product is

u⋆v = B^𝖳((Cu)(Cv)),

where the product inside parentheses is the explicitly defined 63-dimensional Matsuo product. The ideal property in Lemma 2.1 proves that this definition is independent of the chosen lifts.

A second right inverse uses only the 28 difference-root coordinates and makes their quotient classes an explicit basis. Let $𝒞$_diff ⊂ ℚ^$ℛ$ be their coordinate space, and let A_T be the triangular-graph adjacency matrix on two-subsets, joining pairs that meet once. The corresponding square block of B is

$$B_{\mathrm{diff}}=J_{28}-A_T, B_{\mathrm{diff}}^{-1}={\displaystyle \frac{1}{4}}{I}_{28}-{\displaystyle \frac{1}{8}}{A}_T+{\displaystyle \frac{3}{64}}{J}_{28}.$$

(9)

Indeed, A_T has eigenvalues 12,4,−2 with multiplicities 1,7,20: the constants and the seven-dimensional span of centred point-incidence vectors give the first two spaces, and the remaining space has eigenvalue −2. Thus B_diff has eigenvalues 16,−4,2, and multiplication verifies (9). The map $u\mapsto {\iota }_{{\mathcal{C}}_{\mathrm{diff}}}{B}_{\mathrm{diff}}^{-1}u$ is a right inverse of B^T supported on $𝒞$_diff. Hence the difference-root classes form a basis of the quotient. This right inverse depends on the chosen A₇ subsystem; unlike C, it need not be Weyl-equivariant.

The complementary coordinate space on the 35 partition lines is not the radical and does not complement it. If B_P is the submatrix on those lines, then direct column counting gives

$$B_P^{\mathsf{T}}{B}_P=8I_{28}-2A_T+12J_{28}.$$

Its eigenvalues are 320,0,12 on the same three spaces. Hence rankB_P = 21 and the partition-coordinate space meets kerB^T in dimension 35 − 21 = 14. This distinguishes the coordinate split 28 + 35 from the radical filtration.

Appendix C.2 Full Root Frames

In the binary model, Lagrangians become full orthogonal root frames. Proposition 6.4 therefore gives the frame relations for E₇ directly.

Corollary C.2.Differences of sums over full orthogonal E₇root frames span the radical. There are 135 such frames, 15 through each root line.

Proof. The nonzero vectors of a binary Lagrangian correspond to seven orthogonal root lines. These are precisely the full frames. Proposition 6.4 gives the spanning assertion and the count Λ₃ = 135; a point belongs to Λ₂ = 15 frames. □

Appendix C.3 Incidence Supports and the Steiner Design

The columns of the incidence matrices describe root subsystems, while complementary rows describe Steiner complexes. This connects the quotient coordinates with the classical bitangent configuration.

The columns of B = B₋ are indicators of the 28 root subsystems of type E₆ obtained by taking roots perpendicular to a minuscule weight pair. The columns of B⁺=B₊ are indicators of 36 subsystems of type A₇. The standard plus form q₀ selects the 28 difference-root lines. These support identifications can also be checked directly: every minus support contains 72 roots of rank 6, every plus support contains 56 roots of rank 7, and all are closed root subsystems. Exact root enumeration verifies these assertions for all columns.

Proposition C.3 (Steiner complexes).For x ∈ V₆∖{0}, put S_x = {q ∈ Ω₋:q(x) = 0}. The 63 sets S_x form a quasi-symmetric 2-(28,12,11) design. Each is partitioned into six pairs {q,q + b(x,⋅)}. Distinct blocks meet in four elements when b(x,y) = 0 and six when b(x,y) = 1; there are 945 and 1008 unordered pairs of these two kinds, respectively. Under the classical identification of odd theta characteristics with the bitangents of a smooth plane quartic, these are its Steiner complexes [11; 3, Section 3.4].

Proof. The complementary incidence matrix B₀ = J − B has row sum 12, and

$$B_0^T{B}_0=16I+11J, B_0{B}_0^T=8I+2A+4J.$$

The first identity gives replication number 27 and pair multiplicity 11; the second gives the two block intersections. There are 63 ⋅ 32/2 = 1008 adjacent pairs of roots and $\left({\displaystyle \genfrac{}{}{0pt}{}{63}{2}}\right)-1008=945$ nonadjacent pairs. If q(x) = 0, the form q + b(x,⋅) has the same Arf invariant as q. Translation by b(x,⋅) is a fixed-point-free involution of S_x, giving its six pairs. □

For comparison with the local relations in Theorem 4.1, exact enumeration finds 315 subsystems of type D₄. Each has three orthogonal four-frames. Differences of the corresponding frame sums lie in the radical and have exact rational span of dimension 35, so they give another generating family for the positive radical.

Appendix C.4 Equiangular Projector Coordinates and Their Product

The minuscule weight lines provide another basis of the Jordan algebra, with a product determined by the signs of their triple inner products. Retain ${\lambda }_p=\sum _{i\in p}{e}_i-s/4$ for the 28 two-subsets p ⊆ {1,…,8}, and define

$$Q_p={\displaystyle \frac{2}{3}}{\lambda }_p{\lambda }_p^T{|}_{H_7}.$$

These are orthogonal rank-one projections onto the 28 equiangular lines. They satisfy

$$\sum _p{Q}_p=4I, \mathrm{tr}\left(Q_p{Q}_q\right)={\displaystyle \frac{1}{9}} \left(p\ne q\right).$$

Their trace Gram matrix and its inverse are

$$K={\displaystyle \frac{8}{9}}I+{\displaystyle \frac{1}{9}}J, K^{-1}={\displaystyle \frac{9}{8}}I-{\displaystyle \frac{1}{32}}J.$$

The invertible Gram matrix shows that these projectors form a basis of the 28-dimensional Jordan algebra. The underlying equiangular line set and its two-graph are classical [31, 35].

Proposition C.4 (axis and product coordinates).Every root-axis image has the expansion

$$P_{\alpha }={\displaystyle \frac{1}{4}}\sum _{\alpha \cdot {\lambda }_p\ne 0}{Q}_p-{\displaystyle \frac{1}{8}}\sum _{\alpha \cdot {\lambda }_p=0}{Q}_p.$$

Equivalently, for incidence coordinates u = B^Tx, the coefficients of $\Psi \left(x\right)=\sum _p{c}_p{Q}_p$ are

$$c=-{\displaystyle \frac{3}{8}}u+{\displaystyle \frac{1}{64}}\left(1^{T}u\right)1.$$

For distinct p,q, define τ_pqr = (λ_p ⋅ λ_q)(λ_q ⋅ λ_r)(λ_r ⋅ λ_p). Then

$$Q_p\circ Q_q={\displaystyle \frac{1}{9}}\left(Q_p+Q_q\right)+{\displaystyle \frac{1}{36}}\sum _{\begin{array}{c}r\notin \{p,q\}\\ {\tau }_{\mathit{pqr}}>0\end{array}}{Q}_r-{\displaystyle \frac{1}{18}}\sum _{\begin{array}{c}r\notin \{p,q\}\\ {\tau }_{\mathit{pqr}}<0\end{array}}{Q}_r.$$

There are 16 positive-sign indices and 10 negative-sign indices in these sums. The Seidel matrix defined by S_pp = 0 and S_pq = sign(λ_p ⋅ λ_q) for p ≠ q has spectrum 9⁽⁷⁾,(−3)⁽²¹⁾.

Proof. Directly, λ_p ⋅ λ_q = ∣p ∩ q∣ − 1/2. Also

$$\sum _p{\lambda }_p{\lambda }_p^T=6\left(I-{\displaystyle \frac{1}{8}}J\right),$$

which gives the tight-frame identity on H₇. These formulas prove the trace Gram identities. For a root α, the inner products α ⋅ λ_p are 0 or ±1, so tr(P_αQ_p) is 0 or 1/3. Their sum is 4. Multiplication by K⁻¹ therefore gives coefficients −1/8 or 1/4, respectively. Summing the axis expansions and using 1^Tu = 161^Tx gives the coordinate transformation.

For M = Q_p∘Q_q, put h_r = tr(Q_rM). If r = p or q, then h_r = 1/9; for distinct p,q,r, it is sign(τ_pqr)/27. Moreover $\sum _r{h}_r=4\mathrm{tr}M=4/9$. Thus the coefficient of Q_r is 9h_r/8 − 1/72, giving exactly the three values in the displayed product. Taking traces shows that the positive and negative counts satisfy a + b = 26 and 2/9 + a/36 − b/18 = 1/9, hence (a,b) = (16,10). Finally the Gram matrix of the unit vectors $\sqrt{2/3}{\lambda }_p$ is I + S/3. The tight-frame identity gives eigenvalues 4 with multiplicity 7 and 0 with multiplicity 21, proving the Seidel spectrum. □

The sign of τ_pqr does not depend on the choices of signs of the three vectors. It also identifies the classical triad type. With q_p = q₀ + b(p,⋅) as in the binary model,

$$\mathrm{sign}\left({\tau }_{\mathit{pqr}}\right)={\left(-1\right)}^{\mathrm{Arf}\left(q_p+q_q+q_r\right)}.$$

Indeed, sign(λ_p ⋅ λ_q) = (−1)^1+b(p,q) for distinct two-subsets, while

Arf(q_p + q_q + q_r) = q₀(p + q + r) = 1 + b(p,q) + b(q,r) + b(r,p) (mod 2).

A triple of odd characteristics is syzygetic precisely when its sum is odd [3, Definition 3.5]. Thus the 16 positive-sign triples through a pair are azygetic and the 10 negative-sign triples are syzygetic.

Computational Verification: The supplementary archive contains the 25 verification scripts, exact coordinate and rank calculations, logs, and figure sources. Running the suite requires Python 3.11 or later and the dependencies listed in its README. The archive includes the exhaustive Schreier calculation of the ASL(2,3) frame action and the spread enumeration in Proposition 4.5. Checks on small coordinate systems supplement the uniform arguments; the proofs for the infinite families and Propositions 8.1–8.2 do not depend on bounded enumeration.

Use of Generative AI: ChatGPT (OpenAI) and Claude (Anthropic) assisted with mathematical checks, verification-code development and revision of the manuscript. Responsibility for the results and final text rests with the author.