Generation sl3 in e7

Statement

Fix a good embedding $\mathfrak g_{\mathrm{SM}}\subset\mathfrak e_7$ . Its centralizer satisfies

C_{\mathfrak e_7}(\mathfrak g_{\mathrm{SM}}) \cong \mathfrak{sl}_3(\mathbb C)\oplus\mathbb C^2,

and it contains exactly one Lie subalgebra isomorphic to $\mathfrak{sl}_3(\mathbb C)$ . This unique subalgebra is the generation $\mathfrak{sl}_3$ , denoted $\mathfrak{sl}_3^{\mathrm{gen}}$ .

The displayed formula is an isomorphism of Lie algebras: the two summands commute, and $\mathbb C^2$ is abelian.

Why the sl3 is unique

For any subalgebra $\mathfrak a\cong\mathfrak{sl}_3$ inside $\mathfrak{sl}_3\oplus\mathbb C^2$ , projection to the abelian factor has abelian image. Since $\mathfrak{sl}_3$ is simple and perfect, that projection vanishes. Thus $\mathfrak a$ lies in the first summand, and equality follows from dimension.

Root-system description

For a compatible Cartan description of the regular $\mathfrak g_{\mathrm{SM}}$ , the regular-centralizer formula selects the roots orthogonal to the Standard Model root subsystem. They form a subsystem of type $A_2$ , producing $\mathfrak{sl}_3^{\mathrm{gen}}$ ; the remaining two orthogonal Cartan directions give the abelian factor.

Intrinsic status

Once the particular good embedded subalgebra $\mathfrak g_{\mathrm{SM}}\subset\mathfrak e_7$ is fixed, $\mathfrak{sl}_3^{\mathrm{gen}}$ is intrinsic: no Cartan subalgebra or root labeling is required to characterize it. A Cartan choice is needed only to split it into the generation plane and six root spaces or to name the three generation $\mathfrak{sl}_2$ subalgebras.

References

John C. Baez, “Three Generations in E7,” 2026, Proposition 1. arXiv:2608.06271.
Benjamin Nasmith, “An Exceptional Combinatorial Sequence and Standard Model Particles,” 2020, §6. arXiv:2012.03933.
E. B. Dynkin, “Semisimple Subalgebras of Semisimple Lie Algebras,” American Mathematical Society Translations, Series 2, 6 (1957), 111–244.