Three D6 subsystems in E7

Statement

Use the generation-plane partition $\Phi=A\sqcup\Phi_0\sqcup\Phi_1\sqcup\Phi_2\sqcup\Phi_3$ and roots $\pm\beta_k$ defining the three generation $\mathfrak{sl}_2$ 's. For each $k$ ,

\{r\in\Phi:r\perp\beta_k\}=\Phi_0\sqcup\Phi_k

is a rank-six root subsystem with $60$ roots, hence is of type $D_6$ . Its regular Lie subalgebra is denoted

\mathfrak{so}_{12}(\beta_k)\subset\mathfrak e_7.

The three $D_6$ subsystems, and therefore the three embedded copies of $\mathfrak{so}_{12}(\mathbb C)$ , are distinct.

Orthogonality test

Projection to the generation plane reduces the condition $r\perp\beta_k$ to $\pi(r)\perp\beta_k$ . In the projection trichotomy, zero and $\pm w_k$ are orthogonal to $\beta_k$ ; the other four defining weights and all six $A_2$ roots are not. This gives exactly $\Phi_0\sqcup\Phi_k$ .

Centralizer statement

The regular subalgebras

\mathfrak{sl}_2(\beta_k) \quad\text{and}\quad \mathfrak{so}_{12}(\beta_k)

are mutual centralizers in $\mathfrak e_7$ . Their sum is therefore a maximal-rank embedded Lie subalgebra of type $A_1+D_6$ . The associated branching rule describes the complementary $64$ -dimensional module.

Dependence on choices

The unordered triple depends on a Cartan choice in $\mathfrak{sl}_3^{\mathrm{gen}}$ ; labels depend on an ordering of its three root lines. The notation $\mathfrak{so}_{12}(\beta_k)$ is insensitive to changing $\beta_k$ to $-\beta_k$ .

References

John C. Baez, “Three Generations in E7,” 2026, Lemma 4 and Proposition 5. arXiv:2608.06271.
E. B. Dynkin, “Semisimple Subalgebras of Semisimple Lie Algebras,” American Mathematical Society Translations, Series 2, 6 (1957), 111–244.