Standard sl6 in e7 - Knowlpedia

Statement

Fix a good embedded $\mathfrak g_{\mathrm{SM}}\subset\mathfrak e_7$ and let $\mathfrak{sl}_3^{\mathrm{gen}}$ be its generation $\mathfrak{sl}_3$ . Then

C_{\mathfrak e_7}(\mathfrak{sl}_3^{\mathrm{gen}}) \cong\mathfrak{sl}_6(\mathbb C).

This embedded algebra is the standard $\mathfrak{sl}_6$ , denoted $\mathfrak{sl}_6^{\mathrm{SM}}$ . Moreover,

C_{\mathfrak e_7}(\mathfrak{sl}_6^{\mathrm{SM}}) =\mathfrak{sl}_3^{\mathrm{gen}},

so the two are mutual centralizers, and

\mathfrak{sl}_3^{\mathrm{gen}}\oplus\mathfrak{sl}_6^{\mathrm{SM}} \subset\mathfrak e_7

is a maximal embedded Lie subalgebra of type $A_2+A_5$ .

Root-system description

For a compatible generation plane $P$ , let

\Phi_0=\{r\in\Phi(E_7):r\perp P\}.

The projection analysis shows that $\Phi_0$ has rank $5$ and $30$ roots, hence is of type $A_5$ . Its Cartan subspace is $P^\perp$ , so there is no additional abelian centralizer summand; the associated regular algebra is $\mathfrak{sl}_6$ .

Intrinsic status

Although the root-system proof uses a compatible Cartan subalgebra, $\mathfrak{sl}_6^{\mathrm{SM}}$ is intrinsically characterized as the centralizer of $\mathfrak{sl}_3^{\mathrm{gen}}$ . Thus it depends only on the chosen good embedded Standard Model algebra, not on the later choice that labels three generation root lines.

Branching

Restriction of the adjoint $\mathfrak e_7$ -module to this maximal $A_2+A_5$ subalgebra gives the $A_2+A_5$ branching rule. The direct sum $\mathfrak{sl}_3^{\mathrm{gen}}\oplus\mathfrak{sl}_6^{\mathrm{SM}}$ is a Lie-algebra direct sum; the other branching summands are modules, not Lie subalgebras.

References

John C. Baez, “Three Generations in E7,” 2026, Proposition 6. arXiv:2608.06271.
E. B. Dynkin, “Semisimple Subalgebras of Semisimple Lie Algebras,” American Mathematical Society Translations, Series 2, 6 (1957), 111–244, especially the maximal-subalgebra tables.