E7 root-projection trichotomy

Statement

Let $\Phi\subset V$ be the $E_7$ root system, let $A\subset\Phi$ be the $A_2$ root system of the generation $\mathfrak{sl}_3$ , let $P=\operatorname{span}_{\mathbb R}A$ , and let $\pi:V\to P$ be orthogonal projection. If $w_1,w_2,w_3$ are the defining $A_2$ weights, then every $r\in\Phi$ satisfies

\pi(r)\in \{0\}\;\sqcup\; \{\pm w_1,\pm w_2,\pm w_3\}\;\sqcup\;A.

Moreover,

\pi(r)\in A\quad\Longleftrightarrow\quad r\in A.

Why only these projections occur

For every $\delta\in A$ , integrality of Cartan integers gives $\langle\pi(r),\delta\rangle=\langle r,\delta\rangle\in\mathbb Z$ . Hence $\pi(r)$ lies in the $A_2$ weight lattice. With $E_7$ roots normalized to squared length $2$ , orthogonal projection gives $\lVert\pi(r)\rVert^2\le 2$ . The weight-lattice vectors within this bound are precisely zero, the six vectors $\pm w_i$ of squared length $2/3$ , and the six roots of squared length $2$ .

Equality of lengths forces $r=\pi(r)\in P$ . The only rank-two simply laced root subsystem containing $A_2$ is $A_2$ , proving the final equivalence.

The induced partition

Define

\Phi_0=\{r\in\Phi:\pi(r)=0\}, \qquad \Phi_k=\{r\in\Phi:\pi(r)=\pm w_k\}.

Then

\Phi=A\sqcup\Phi_0\sqcup\Phi_1\sqcup\Phi_2\sqcup\Phi_3,

with $|\Phi_0|=|\Phi_1|=|\Phi_2|=|\Phi_3|=30$ . The set $\Phi_0$ is an $A_5$ root subsystem; the sets $\Phi_k$ are not root systems but index the 30-dimensional generation modules.

Dependence on choices

The trichotomy is valid for any compatible generation-plane choice. The sets $\Phi_k$ are permuted when the weights $w_k$ are relabeled; no ordering of the three sets is intrinsic to the good Standard Model embedding alone.

References

John C. Baez, “Three Generations in E7,” 2026, Lemmas 2–3. arXiv:2608.06271.
Benjamin Nasmith, “An Exceptional Combinatorial Sequence and Standard Model Particles,” 2020, Lemma 2.1. arXiv:2012.03933.
P. J. Cameron, J. M. Goethals, J. J. Seidel, and E. E. Shult, “Line Graphs, Root Systems, and Elliptic Geometry,” Journal of Algebra 43 (1976), 305–327. DOI record90162-9).