Orthogonal Frames and Maximal Orthogonal Subsystems
in the Exceptional Root Systems $E_6$ , $E_7$ , and $E_8$

October 2026

Abstract

We study ordered orthogonal frames of roots and maximal orthogonal subsystems in the exceptional simply-laced root systems $E_6$ , $E_7$ , and $E_8$ . The central structural phenomenon is that sufficiently large orthogonal frames exhibit unique completion rigidity.

For $E_7$ , we construct a canonical projection from the root lattice modulo $2$ onto a six-dimensional symplectic vector space over $\mathbb F_2$ . This map identifies the $63$ root lines with the nonzero vectors of $\mathbb F_2^6$ , and Euclidean orthogonality with symplectic orthogonality. Consequently, maximal orthogonal subsystems of type $A_1^7$ correspond bijectively to Lagrangian subspaces, of which there are exactly $135$ . By a dimension-cardinality argument, any orthogonal set of $k\geq4$ root lines spans a full $3$ -dimensional Lagrangian subspace, establishing that every orthogonal frame of size at least four has a unique maximal completion in $E_7$ .

For $E_8$ , the orthogonal complement of every root line is isomorphic to $E_7$ . Slicing and incidence double-counting prove that $E_8$ contains exactly $2025$ maximal subsystems of type $A_1^8$ . By reducing intersections through a common root line to the $E_7$ Lagrangian geometry, we prove that any two distinct $A_1^8$ subsystems share at most four root lines. This establishes that every orthogonal frame of size at least five has a unique maximal $A_1^8$ completion in $E_8$ .

These completion theorems yield exact, closed-form enumerations for all higher frame counts: $N_k(E_7)=135\cdot2^k\cdot7^{\underline{k}} \qquad (4\leq k\leq7)$ and $N_k(E_8)=2025\cdot2^k\cdot8^{\underline{k}} \qquad (5\leq k\leq8).$ Furthermore, single-root slicing establishes the exact reduction $N_k(E_8)=240\,N_{k-1}(E_7) \qquad (1\leq k\leq8).$ We show that the two closed expressions agree for every $k\in\{5,6,7,8\}$ , and that this agreement reflects the geometric compatibility between the ambient $A_1^8$ cross-polytopal decomposition of $E_8$ and its $E_7$ root-complement slicing.

Introduction

Background and motivation

Let $R$ be an irreducible, simply-laced root system in a Euclidean vector space $(E,(\cdot,\cdot))$ , normalized so that $(r,r)=2 \qquad\text{for all }r\in R.$ An ordered orthogonal $k$ -frame in $R$ is an ordered sequence $(r_1,\ldots,r_k)\in R^k$ satisfying $(r_i,r_j)=2\delta_{ij}.$ We denote by $N_k(R)$ the number of such ordered orthogonal $k$ -frames.

For the infinite classical families $A_n$ and $D_n$ , uniform combinatorial formulas for $N_k(R)$ are well understood. In contrast, for the exceptional root systems $E_6$ , $E_7$ , and $E_8$ , one naturally encounters a sequence of changing orthogonal complements. This permits direct computation of small frame counts, but obscures the global geometry of large orthogonal systems.

The purpose of this paper is to make the high-order structure explicit. The main phenomenon is a rigidity statement: sufficiently large orthogonal systems determine their maximal orthogonal completion uniquely.

Summary of results

To avoid ambiguities involving signs, we work primarily with root lines $L=\{\pm r\}.$ These correspond to one-dimensional root sublattices of type $A_1$ . A maximal orthogonal collection of root lines therefore determines a subsystem of type $A_1^m$ .

The main results are as follows.

The symplectic model of $E_7$ . We construct a canonical quotient $Q(E_7)/2Q(E_7)\longrightarrow \mathbb F_2^6$ whose kernel is the radical of the induced alternating form. The resulting symplectic space identifies the $63$ root lines of $E_7$ with the $63$ nonzero vectors of $\mathbb F_2^6$ .
Lagrangian classification. The $A_1^7$ subsystems of $E_7$ correspond bijectively to the $3$ -dimensional totally isotropic subspaces of $\mathbb F_2^6$ . There are exactly $135$ such Lagrangians.
Unique completion in $E_7$ . Every orthogonal set of at least four root lines spans a Lagrangian, and hence determines a unique $A_1^7$ subsystem.
$E_8$ slicing. The orthogonal complement of every root line in $E_8$ is an $E_7$ root system. Incidence double-counting therefore gives exactly $2025$ maximal $A_1^8$ subsystems in $E_8$ .
Unique completion in $E_8$ . Any two distinct $A_1^8$ subsystems share at most four root lines. Consequently, every orthogonal system of at least five root lines is contained in a unique $A_1^8$ subsystem.
Closed frame formulas. For the high-order ranges, $N_k(E_7)=135\cdot2^k\cdot7^{\underline{k}} \qquad(4\leq k\leq7),$ and $N_k(E_8)=2025\cdot2^k\cdot8^{\underline{k}} \qquad(5\leq k\leq8).$
Compatibility with slicing. For every $k\in\{5,6,7,8\},$ the two descriptions of $N_k(E_8)$ agree: $N_k(E_8)=240\,N_{k-1}(E_7) =2025\cdot2^k\cdot8^{\underline{k}}.$

Distinction from standard background

We separate standard Lie-theoretic facts from the arguments developed here.

Classical background: the root counts $|E_6|=72$ , $|E_7|=126$ , $|E_8|=240$ ; the root-complement identifications; the standard Weyl-group facts; the isomorphism $W(E_7)/\{\pm I\}\cong\operatorname{Sp}(6,\mathbb F_2);$ and the existence of the maximal $A_1^7$ and $A_1^8$ systems.
Arguments developed here: the root-lattice construction of the symplectic quotient; the dimension-cardinality proof of unique completion in $E_7$ ; the incidence derivation of the $2025$ count in $E_8$ ; the intersection bound for $A_1^8$ systems; the resulting unique completion theorem in $E_8$ ; and the closed high-order frame formulas.

Root lines and frame enumeration

Definition 1. Let $R$ be a simply-laced root system. A root line of $R$ is a pair of opposite roots $L=\{\pm r\}\subset R.$ We denote the set of root lines by $\mathcal L(R)$ . Thus $|\mathcal L(R)|=\frac{|R|}{2}.$

Two root lines $L_1=\{\pm r_1\}, \qquad L_2=\{\pm r_2\}$ are orthogonal if $(r_1,r_2)=0.$ This is independent of the choices of representatives.

Definition 2. An orthogonal $k$ -line system is a set $\{L_1,\ldots,L_k\}\subseteq\mathcal L(R)$ of $k$ pairwise orthogonal root lines. We denote the number of unordered orthogonal $k$ -line systems by $M_k(R)$ .

Lemma 3. For every simply-laced root system $R$ , $N_k(R)=2^k k!M_k(R),$ or equivalently, $M_k(R)=\frac{N_k(R)}{2^k k!}.$

Proof. Given an unordered set of $k$ mutually orthogonal root lines, there are $k!$ ways to order them and two choices of sign for each line. Thus every unordered system produces exactly $2^k k!$ ordered frames. ◻

Definition 4. A root subsystem $S\subseteq R$ is of type $A_1^m$ if it consists of $m$ mutually orthogonal root lines together with their opposite roots.

If $R$ has rank $n$ , then $M_k(R)=N_k(R)=0 \qquad(k>n).$

The complete sequence for $E_6$

Lemma 5. The maximum size of an orthogonal set of root lines in $E_6$ is $4$ .

Proof. Suppose there were five mutually orthogonal root lines in $E_6$ . They would generate an $A_1^5$ subsystem of rank $5$ . The classification of rank- $5$ subsystems in $E_6$ gives types $A_5$ , $D_5$ , and $A_4\oplus A_1$ , none of which contains $A_1^5$ . Thus no such set exists. ◻

Proposition 6. For $1\leq k\leq4$ , $N_k(E_6)=\prod_{j=1}^k d_j, \qquad (d_1,d_2,d_3,d_4)=(72,30,12,2).$ Hence $$\begin{array}{c|ccccc} \toprule k&1&2&3&4&\geq5\\ \midrule N_k(E_6)&72&2\,160&25\,920&51\,840&0\\ M_k(E_6)&36&270&540&135&0\\ \bottomrule \end{array}$$

Proof. There are $72$ choices for the first root. The orthogonal complement of a root in $E_6$ is $A_5$ , containing $30$ roots.

Inside $A_5$ , the orthogonal complement of a root is $A_3$ , containing $12$ roots. The orthogonal complement of a root in $A_3$ is $A_1$ , containing $2$ roots. Thus $N_1=72,\quad N_2=72\cdot30,\quad N_3=72\cdot30\cdot12,\quad N_4=72\cdot30\cdot12\cdot2.$ The values of $M_k$ follow from 3. ◻

Remark 7. The value $M_4(E_6)=135$ counts the maximal $A_1^4$ systems in $E_6$ .

The symplectic model of $E_7$

The exceptional root system $E_7$ has $126$ roots and therefore $63$ root lines. We now construct its symplectic model directly from the root lattice.

Construction from the root lattice modulo $2$

Let $Q=Q(E_7)=\bigoplus_{i=1}^7\mathbb Z\alpha_i$ be the root lattice. Let $P=Q^*$ be its dual lattice. Since $[P:Q]=2,$ we have $\det(Q)=2.$

Set $\overline Q=Q/2Q\cong\mathbb F_2^7.$ The root-lattice form induces $\overline b:\overline Q\times\overline Q\longrightarrow\mathbb F_2, \qquad \overline b(x+2Q,y+2Q)=(x,y)\pmod2.$

Lemma 8. The form $\overline b$ is alternating and has a one-dimensional radical.

Proof. Since $Q$ is even, $(x,x)\in2\mathbb Z$ for all $x\in Q$ . Hence $\overline b(u,u)=0$ for all $u\in\overline Q$ , so $\overline b$ is alternating.

Now $x+2Q$ belongs to the radical precisely when $(x,y)\equiv0\pmod2$ for all $y\in Q$ . Equivalently, $x/2\in Q^*=P.$ Thus $\operatorname{Rad}(\overline b) = (2P\cap Q)/2Q = 2P/2Q \cong P/Q.$ Since $[P:Q]=2$ , the radical has dimension one over $\mathbb F_2$ . ◻

Let $u_0\in\overline Q\setminus\{0\}$ denote the unique nonzero radical vector. In terms of fundamental weights, one may take $u_0=2\varpi_7+2Q.$

Definition 9. The canonical symplectic space of $E_7$ is $V=\overline Q/\operatorname{Rad}(\overline b).$ Thus $V\cong\mathbb F_2^6.$ The form $\overline b$ descends to a nondegenerate alternating form $\omega:V\times V\longrightarrow\mathbb F_2.$

Consequently, $(V,\omega)\cong(\mathbb F_2^6,\omega)$ is a six-dimensional symplectic vector space.

The symplectic group has order $|\operatorname{Sp}(6,\mathbb F_2)| = 2^9(2^2-1)(2^4-1)(2^6-1) = 1\,451\,520.$ The Weyl group satisfies $|W(E_7)|=2\,903\,040$ and classically $W(E_7)/\{\pm I\}\cong\operatorname{Sp}(6,\mathbb F_2).$

The root-line bijection

Let $\pi:Q\longrightarrow V$ be the quotient map $x\longmapsto(x+2Q)+\operatorname{Rad}(\overline b).$

Proposition 10. The map $\pi$ has the following properties.

For every root $r\in E_7$ , $\pi(r)\neq0.$
For roots $r_1,r_2\in E_7$ , $\pi(r_1)=\pi(r_2) \quad\Longleftrightarrow\quad r_2=\pm r_1.$
The induced map $\phi:\mathcal L(E_7)\longrightarrow V\setminus\{0\}$ is a bijection.
For root lines $L_1,L_2$ , $L_1\perp L_2 \quad\Longleftrightarrow\quad \omega(\phi(L_1),\phi(L_2))=0.$

Proof. For (i), suppose $\pi(r)=0$ . Then $r+2Q\in\operatorname{Rad}(\overline b).$ Since $r\neq0$ , this gives $r\in2P.$ Hence $(r,\alpha)\equiv0\pmod2$ for every root $\alpha$ . But every root in an irreducible simply-laced root system of rank at least $2$ has a root with inner product $\pm1$ . This is a contradiction.

For (ii), suppose $\pi(r_1)=\pi(r_2).$ Then $r_1-r_2\in\operatorname{Rad}(\overline b).$ By definition of the radical, $(r_1-r_2,\alpha)\equiv0\pmod2 \qquad \text{for every root }\alpha\in E_7.$ Taking $\alpha=r_1$ gives $2-(r_1,r_2)\equiv0\pmod2.$ Hence $(r_1,r_2)$ is even. Since the inner products of two roots in a simply-laced system belong to $\{0,\pm1,\pm2\},$ we obtain $(r_1,r_2)\in\{0,\pm2\}.$

If $(r_1,r_2)=\pm2,$ then equality holds in the Cauchy–Schwarz inequality, so $r_2=\pm r_1.$

It remains to exclude $(r_1,r_2)=0.$ In that case $r_1\in R_2:=r_2^\perp\cap E_7.$ The orthogonal complement of a root in $E_7$ is a root system of type $D_6$ . Since $D_6$ is irreducible of rank at least two, every root of $D_6$ has a root adjacent to it, i.e., there exists $\alpha\in R_2$ such that $(r_1,\alpha)=\pm1.$ Because $\alpha\in r_2^\perp$ , $(r_2,\alpha)=0.$ Therefore $(r_1-r_2,\alpha) = (r_1,\alpha)-(r_2,\alpha) = \pm1,$ which is odd, contradicting $(r_1-r_2,\alpha)\equiv0\pmod2.$ Thus $(r_1,r_2)=0$ is impossible, and $r_2=\pm r_1.$

For (iii), there are $|\mathcal L(E_7)|=63$ root lines and $|V\setminus\{0\}|=2^6-1=63$ nonzero vectors. Part (ii) gives injectivity, hence bijectivity.

For (iv), $\omega(\phi(L_1),\phi(L_2)) \equiv(r_1,r_2)\pmod2.$ For distinct root lines, the inner product cannot be $\pm2$ , so it belongs to $\{0,\pm1\}$ . Hence it is even precisely when it is zero. Thus $L_1\perp L_2 \quad\Longleftrightarrow\quad \omega(\phi(L_1),\phi(L_2))=0.$ ◻

Lagrangian subspaces and the $135$ maximal systems

Definition 11. A subspace $W\subseteq V$ is totally isotropic if $\omega(u,v)=0 \qquad \text{for all }u,v\in W.$ Since $\dim V=6$ , the largest possible dimension of a totally isotropic subspace is $3$ . A $3$ -dimensional totally isotropic subspace is called a Lagrangian.

Proposition 12. Under the bijection $\phi$ , maximal orthogonal sets of root lines in $E_7$ correspond bijectively to Lagrangian subspaces of $V$ . Consequently every maximal orthogonal subsystem is of type $A_1^7$ .

Proof. Let $W$ be Lagrangian. It contains $2^3-1=7$ nonzero vectors. By 10, these correspond to seven pairwise orthogonal root lines.

Conversely, let $\mathcal S$ be a maximal orthogonal set of root lines. Its image under $\phi$ is pairwise symplectically orthogonal. Let $W=\operatorname{Span}_{\mathbb F_2}(\phi(\mathcal S)).$ Then $W$ is totally isotropic, so $\dim W\leq3.$ If $\dim W<3$ , then $W$ lies in a Lagrangian $W'$ and $\phi^{-1}(W'\setminus\{0\})$ would be a strictly larger orthogonal system. Thus $\dim W=3.$ Hence $W$ is Lagrangian and contains exactly seven nonzero vectors. ◻

Theorem 13. There are exactly $135$ maximal $A_1^7$ subsystems in $E_7$ .

Proof. Count ordered bases of Lagrangian subspaces.

Choose an ordered isotropic basis $(u_1,u_2,u_3).$ There are $2^6-1=63$ choices for $u_1$ .

Once $u_1$ is chosen, $u_2$ must lie in $u_1^\perp$ but outside $\operatorname{Span}(u_1)$ , giving $2^5-2=30$ choices.

Finally, $u_3$ must lie in $\operatorname{Span}(u_1,u_2)^\perp$ but outside $\operatorname{Span}(u_1,u_2)$ , giving $2^4-2^2=12$ choices.

Thus there are $63\cdot30\cdot12$ ordered isotropic bases.

Each Lagrangian has $|\operatorname{GL}_3(\mathbb F_2)| = (8-1)(8-2)(8-4) = 168$ ordered bases. Therefore $|\operatorname{Lag}(V)| = \frac{63\cdot30\cdot12}{168} = 135.$ ◻

Intersections of $A_1^7$ subsystems

Theorem 14. Let $T_1,T_2$ be distinct $A_1^7$ subsystems of $E_7$ . Then $|T_1\cap T_2|\leq3.$ More precisely, $|T_1\cap T_2|\in\{0,1,3\}.$

Proof. Let $W_1,W_2$ be the corresponding Lagrangians. Since they are distinct, $d:=\dim(W_1\cap W_2)\leq2.$ The common root lines correspond to the nonzero vectors of $W_1\cap W_2$ . Hence $|T_1\cap T_2| = 2^d-1.$ For $d=0,1,2$ this gives $0,\quad1,\quad3.$ ◻

Remark 15. The Lagrangians are Fano planes $PG(2,\mathbb F_2)$ inside the symplectic polar space $W(5,2)$ . Two distinct Lagrangians can therefore meet in the empty set, a point, or a projective line containing three points. In particular, intersection size $2$ cannot occur.

Unique completion in $E_7$

Theorem 16 (Unique completion in $E_7$ ). Let $\mathcal F=\{L_1,\ldots,L_k\}$ be an orthogonal set of root lines in $E_7$ . If $4\leq k\leq7,$ then $\mathcal F$ is contained in a unique $A_1^7$ subsystem.

Proof. Set $S=\phi(\mathcal F)$ and $W=\operatorname{Span}_{\mathbb F_2}(S).$ Since $S$ is pairwise symplectically orthogonal, $W$ is totally isotropic and therefore $\dim W\leq3.$ But $W$ contains at least four distinct nonzero vectors. A two-dimensional vector space over $\mathbb F_2$ has only $2^2-1=3$ nonzero vectors. Thus $\dim W=3.$ Hence $W$ is Lagrangian.

Any $A_1^7$ subsystem containing $\mathcal F$ corresponds to a Lagrangian $W_T$ containing $S$ . Since $W_T$ is a vector space, $W\subseteq W_T.$ Both spaces have dimension $3$ , so $W=W_T.$ Thus the completion is unique. ◻

Corollary 17. For every $k\in\{4,5,6,7\},$ the $135$ maximal $A_1^7$ subsystems partition the set of orthogonal $k$ -line systems in $E_7$ .

Frame enumeration in $E_7$

Theorem 18. For $4\leq k\leq7,$ we have $\boxed{ N_k(E_7) = 135\cdot2^k\cdot7^{\underline{k}} }.$ Equivalently, $\boxed{ M_k(E_7) = 135\binom7k. }$

Proof. Every maximal $A_1^7$ subsystem contains seven mutually orthogonal root lines. By 17, every orthogonal $k$ -line system with $k\geq4$ belongs to exactly one such subsystem. Therefore $M_k(E_7) = 135\binom7k.$ Applying 3, $N_k(E_7) = 2^k k!\,135\binom7k = 135\cdot2^k\cdot7^{\underline{k}}.$ ◻

Complete $E_7$ spectrum

For $k\leq3$ , multiple Lagrangians may contain the same system.

For $k=1$ , $N_1(E_7)=126, \qquad M_1(E_7)=63.$

For $k=2$ , each root has $30$ orthogonal roots, so $M_2(E_7) = \frac{63\cdot30}{2} = 945,$ and hence $N_2(E_7)=945\cdot2^2\cdot2! =7\,560.$

For $k=3$ , there are two types of isotropic triples. Dependent triples have the form $u+v+w=0$ inside a Lagrangian plane. Their number is $\frac{63\cdot30}{6}=315.$ Independent triples are bases of Lagrangians, giving $135\cdot\frac{168}{6}=3\,780.$ Thus $M_3(E_7)=315+3\,780=4\,095$ and $N_3(E_7) = 4\,095\cdot2^3\cdot3! = 196\,560.$

Consequently, $$\begin{array}{c|ccccccc|c} \toprule k&1&2&3&4&5&6&7&\geq8\\ \midrule N_k(E_7) &126 &7\,560 &196\,560 &1\,814\,400 &10\,886\,400 &43\,545\,600 &87\,091\,200 &0 \\ M_k(E_7) &63 &945 &4\,095 &4\,725 &2\,835 &945 &135 &0 \\ \bottomrule \end{array}$$

The $E_8$ slice correspondence

The root system $E_8$ has $|E_8|=240$ roots and $|\mathcal L(E_8)|=120$ root lines.

Lemma 19. For every root line $L\in\mathcal L(E_8)$ , $L^\perp\cap E_8\cong E_7.$

Proof. The Weyl group $W(E_8)$ acts transitively on roots. It is therefore enough to consider one root.

In the standard realization of $E_8$ , choose $r=(1,1,0,0,0,0,0,0).$ The roots orthogonal to $r$ form a rank- $7$ root system. The standard root-complement classification identifies it as $E_7$ . Since $E_8$ has $240$ roots, the complement has $126$ roots, as required for $E_7$ . ◻

Counting maximal $A_1^8$ subsystems in $E_8$

Theorem 20. The root system $E_8$ contains exactly $2025$ maximal subsystems of type $A_1^8$ .

Proof. Consider $\Omega = \{(L,S): L\in\mathcal L(E_8),\ S\text{ is an }A_1^8\text{ subsystem},\ L\subset S\}.$

There are $120$ root lines. Fix one line $L$ . By 19, $L^\perp\cap E_8\cong E_7.$ An $A_1^8$ subsystem containing $L$ is obtained by choosing an $A_1^7$ subsystem in this $E_7$ complement. There are $135$ such systems. Hence $|\Omega| = 120\cdot135 = 16\,200.$

On the other hand, each $A_1^8$ subsystem contains exactly eight root lines, so $|\Omega| = 8\cdot\#\{A_1^8\subset E_8\}.$ Therefore $\#\{A_1^8\subset E_8\} = \frac{16\,200}{8} = 2025.$ ◻

Remark 21. The value $2025=45^2$ also agrees with the classical coding-theoretic description of the $E_8$ lattice using the extended binary Hamming code. The double-counting argument above, however, obtains the number directly from the $E_7$ geometry.

Intersections of maximal $A_1^8$ systems

Theorem 22. Let $S_1,S_2$ be distinct $A_1^8$ subsystems of $E_8$ . Then $\boxed{ |S_1\cap S_2|\leq4. }$

Proof. Suppose $S_1$ and $S_2$ have a common root line $L$ . Then every other common root line lies in $L^\perp\cap E_8\cong E_7.$ Removing the common line $L$ , the two $A_1^8$ systems determine two $A_1^7$ systems in this $E_7$ slice.

If $S_1\neq S_2$ , the two induced $A_1^7$ systems are distinct. By 14, they share at most three additional root lines. Therefore $|S_1\cap S_2| \leq1+3=4.$ If the two systems have no common line, the bound is immediate. ◻

Remark 23. Computational enumeration suggests that intersection size $3$ does not occur among distinct maximal $A_1^8$ systems. This observation is not required for any result in this paper and is not promoted to a theorem. Determining the complete intersection distribution of the $2025$ maximal $A_1^8$ systems is left as an open combinatorial question.

Unique completion in $E_8$

Theorem 24. Let $\mathcal F=\{L_1,\ldots,L_k\}$ be an orthogonal set of root lines in $E_8$ . If $5\leq k\leq8,$ then $\mathcal F$ is contained in a unique $A_1^8$ subsystem.

Proof. Existence follows by extending any orthogonal set to a maximal orthogonal set.

For uniqueness, suppose two distinct $A_1^8$ systems $S_1,S_2$ contain $\mathcal F$ . Then $\mathcal F\subseteq S_1\cap S_2.$ Since $|\mathcal F|\geq5$ , this gives $|S_1\cap S_2|\geq5,$ contradicting 22. Hence the maximal completion is unique. ◻

Corollary 25. For every $k\in\{5,6,7,8\},$ the $2025$ maximal $A_1^8$ subsystems partition the set of orthogonal $k$ -line systems in $E_8$ .

High-order frame enumeration in $E_8$

Theorem 26. For $5\leq k\leq8,$ the number of ordered orthogonal $k$ -frames in $E_8$ is $\boxed{ N_k(E_8) = 2025\cdot2^k\cdot8^{\underline{k}}. }$ Equivalently, $\boxed{ M_k(E_8) = 2025\binom8k. }$

Proof. By 24, every orthogonal $k$ -line system with $k\geq5$ lies in a unique maximal $A_1^8$ system. Therefore $M_k(E_8) = 2025\binom8k.$ Applying 3, $N_k(E_8) = 2^k k!M_k(E_8) = 2025\cdot2^k\cdot8^{\underline{k}}.$ ◻

Single-root slicing and compatibility

Proposition 27. For every $1\leq k\leq8,$ one has $\boxed{ N_k(E_8)=240\,N_{k-1}(E_7). }$

Proof. Choose the first root $r_1$ of an ordered orthogonal $k$ -frame in $E_8$ . There are $240$ choices.

Once $r_1$ is chosen, all remaining roots lie in $r_1^\perp\cap E_8\cong E_7.$ Thus the remaining $k-1$ entries form an ordered orthogonal $(k-1)$ -frame in $E_7$ . Conversely, every such frame together with $r_1$ produces an ordered orthogonal $k$ -frame in $E_8$ .

Hence $N_k(E_8)=240\,N_{k-1}(E_7).$ ◻

Theorem 28. For every $k\in\{5,6,7,8\},$ the slicing formula and the maximal-completion formula agree: $\boxed{ 240\,N_{k-1}(E_7) = 2025\cdot2^k\cdot8^{\underline{k}}. }$

Proof. For $k\in\{5,6,7,8\},$ we have $k-1\in\{4,5,6,7\}.$ Applying the $E_7$ high-order formula gives $N_{k-1}(E_7) = 135\cdot2^{k-1}\cdot7^{\underline{k-1}}.$ Therefore $240\,N_{k-1}(E_7) = (240\cdot135) 2^{k-1}7^{\underline{k-1}} = 32\,400 2^{k-1}7^{\underline{k-1}}.$

On the other hand, $8^{\underline{k}} = 8\cdot7^{\underline{k-1}}$ and $2^k=2\cdot2^{k-1}.$ Thus $2025\cdot2^k\cdot8^{\underline{k}} = 2025\cdot(2\cdot8) 2^{k-1}7^{\underline{k-1}}.$ Since $2025\cdot16 = 32\,400$ and $240\cdot135 = 32\,400,$ the two expressions are equal for every $k\in\{5,6,7,8\}.$ ◻

Remark 29. The equality $240\cdot135=2025\cdot16$ is the numerical compatibility between the $240$ roots of $E_8$ , the $135$ maximal $A_1^7$ systems in an $E_7$ slice, and the $2025$ maximal $A_1^8$ systems in $E_8$ . The equality is not merely an arithmetic coincidence: both sides count the same high-order orthogonal-frame structures through two different geometric decompositions.

The complete $E_8$ spectrum

The low-order values can be obtained directly by slicing.

For $k=1$ , $N_1(E_8)=240, \qquad M_1(E_8)=120.$

For $k=2$ , $N_2(E_8) = 240\cdot126 = 30\,240.$ Hence $M_2(E_8) = \frac{30\,240}{2^2\cdot2!} = 3\,780.$

For $k=3$ , $N_3(E_8) = 240\cdot7\,560 = 1\,814\,400,$ so $M_3(E_8) = 37\,800.$

For $k=4$ , $N_4(E_8) = 240\cdot196\,560 = 47\,174\,400,$ and therefore $M_4(E_8) = 122\,850.$

For $k=5,6,7,8$ , the unique-completion formula gives the remaining values.

Thus the complete $E_8$ spectrum is $$\begin{array}{c|ccccccccc} \toprule k&1&2&3&4&5&6&7&8&\geq9\\ \midrule N_k(E_8) &240 &30\,240 &1\,814\,400 &47\,174\,400 &435\,456\,000 &2\,612\,736\,000 &10\,450\,944\,000 &20\,901\,888\,000 &0 \\ M_k(E_8) &120 &3\,780 &37\,800 &122\,850 &113\,400 &56\,700 &16\,200 &2\,025 &0 \\ \bottomrule \end{array}$$

In particular, the high-order values satisfy $N_k(E_8) = 2025\cdot2^k\cdot8^{\underline{k}}$ for $k\in\{5,6,7,8\},$ and simultaneously $N_k(E_8)=240\,N_{k-1}(E_7).$

Unified master spectrum

For convenient comparison, we collect the complete ordered-frame and root-line spectra of all three exceptional systems.

Complete orthogonal frame spectrum $N_k(R)$ and line spectrum $M_k(R)$ for $E_6$ , $E_7$ , and $E_8$ .
$k$	$N_k(E_6)$	$N_k(E_7)$	$N_k(E_8)$
$1$	$72$	$126$	$240$
$2$	$2\,160$	$7\,560$	$30\,240$
$3$	$25\,920$	$196\,560$	$1\,814\,400$
$4$	$51\,840$	$1\,814\,400$	$47\,174\,400$
$5$	$0$	$10\,886\,400$	$435\,456\,000$
$6$	$0$	$43\,545\,600$	$2\,612\,736\,000$
$7$	$0$	$87\,091\,200$	$10\,450\,944\,000$
$8$	$0$	$0$	$20\,901\,888\,000$
$\geq9$	$0$	$0$	$0$
$k$	$M_k(E_6)$	$M_k(E_7)$	$M_k(E_8)$
$1$	$36$	$63$	$120$
$2$	$270$	$945$	$3\,780$
$3$	$540$	$4\,095$	$37\,800$
$4$	$135$	$4\,725$	$122\,850$
$5$	$0$	$2\,835$	$113\,400$
$6$	$0$	$945$	$56\,700$
$7$	$0$	$135$	$16\,200$
$8$	$0$	$0$	$2\,025$
$\geq9$	$0$	$0$	$0$

The table satisfies throughout the identity $N_k(R)=2^k k!M_k(R).$

Structural interpretation

The preceding calculations exhibit three distinct regimes.

$E_6$

The maximal orthogonal rank is $4$ , despite the ambient rank being $6$ . The complete sequence is therefore obtained by a finite chain of orthogonal complements $E_6\supset A_5\supset A_3\supset A_1.$

$E_7$

The $E_7$ geometry is governed globally by the symplectic space $(\mathbb F_2^6,\omega).$ The $63$ root lines become the $63$ nonzero vectors, and the $A_1^7$ systems become Lagrangian subspaces.

The threshold $k=4$ is explained purely by finite-dimensional linear algebra: four distinct nonzero vectors cannot lie in a two-dimensional $\mathbb F_2$ -space, while every totally isotropic subspace has dimension at most three.

$E_8$

The $E_8$ geometry inherits this structure through root-complement slicing. Every root line determines an $E_7$ slice, and every $A_1^8$ system is obtained by adjoining the distinguished root line to an $A_1^7$ system in that slice.

The threshold $k=5$ follows from the intersection bound $|S_1\cap S_2|\leq4.$

Frame size and Siegel genus

It is important to distinguish two independent parameters that can occur in applications to Siegel modular forms.

The integer $k$ used throughout this paper denotes the frame size, namely the number of mutually orthogonal roots in an ordered orthogonal frame.

By contrast, the integer $g$ denotes the Siegel genus of an ambient Siegel modular form.

These quantities should not be identified. In particular, the appearance of $N_4(R)$ in genus-four modular-form calculations is a statement about the role of four-fold orthogonal configurations in a particular modular-form setting; it does not mean that the frame size $k$ is itself a genus parameter.

The present paper concerns the root-theoretic quantities $N_k(E_6),\qquad N_k(E_7),\qquad N_k(E_8),$ independently of any particular Siegel genus.

Further questions

Several natural questions remain.

Determine the complete intersection distribution of the $2025$ maximal $A_1^8$ subsystems of $E_8$ . The present argument proves only the upper bound $|S_1\cap S_2|\leq4$ for distinct systems. Computational evidence suggests that intersection size $3$ may be absent, but this is not proved here.
Give a direct intrinsic combinatorial model for the $2025$ maximal $A_1^8$ systems analogous to the Lagrangian model for $E_7$ .
Determine whether analogous unique-completion phenomena occur in other exceptional or non-crystallographic root configurations.
Study the incidence structures obtained from the maximal orthogonal subsystems and their intersections, including their association schemes and possible connections with coding theory.

Conclusion

The exceptional systems $E_6$ , $E_7$ , and $E_8$ exhibit markedly different global orthogonal-frame geometries.

For $E_6$ , maximal orthogonal systems have size four and the complete spectrum is obtained by direct orthogonal-complement enumeration.

For $E_7$ , reduction modulo $2$ produces a canonical six-dimensional symplectic space. The $63$ root lines become its nonzero vectors, and maximal orthogonal systems become Lagrangian subspaces. This gives exactly $135$ maximal $A_1^7$ systems and proves unique completion beginning at frame size four.

For $E_8$ , every root complement is $E_7$ . This produces exactly $2025$ maximal $A_1^8$ systems. Their intersection is bounded by four root lines, and hence frame size five is the threshold for unique completion.

The resulting high-order formulas are $N_k(E_7) = 135\cdot2^k\cdot7^{\underline{k}} \qquad (4\leq k\leq7),$ and $N_k(E_8) = 2025\cdot2^k\cdot8^{\underline{k}} \qquad (5\leq k\leq8).$ For $k\in\{5,6,7,8\},$ these agree exactly with the root-slicing identity $N_k(E_8)=240\,N_{k-1}(E_7).$

Thus the enumeration of high-order orthogonal frames is controlled not merely by successive branching of root systems, but by a rigid finite-geometric completion structure.

N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Springer, Berlin, 2002.

R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer, New York, 1999.

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. Ser. 2 6 (1957), 111–244.

Introduction

Background and motivation

Summary of results

Distinction from standard background

Root lines and frame enumeration

The complete sequence for E6E_6

The symplectic model of E7E_7

Construction from the root lattice modulo 22

The root-line bijection

Lagrangian subspaces and the 135135 maximal systems

Intersections of A17A_1^7 subsystems

Unique completion in E7E_7

Frame enumeration in E7E_7

Complete E7E_7 spectrum

The E8E_8 slice correspondence

Counting maximal A18A_1^8 subsystems in E8E_8

Intersections of maximal A18A_1^8 systems

Unique completion in E8E_8

High-order frame enumeration in E8E_8