Good Standard Model embedding in e7

Definition

Let

\mathfrak g_{\mathrm{SM}}\cong \mathfrak{sl}_3(\mathbb C)\oplus\mathfrak{sl}_2(\mathbb C)\oplus\mathbb C

be the complexified Standard Model Lie algebra. A good Standard Model embedding in $\mathfrak e_7$ is an embedding $\iota:\mathfrak g_{\mathrm{SM}}\hookrightarrow\mathfrak e_7$ carried by an automorphism of $\mathfrak e_7$ to the regular embedding obtained by successively removing simple roots down the exceptional chain and then taking the relevant maximal Levi subalgebra of $\mathfrak{sl}_5$ .

Thus “good” specifies an automorphism class of embeddings, not merely an abstract subalgebra isomorphic to $\mathfrak g_{\mathrm{SM}}$ .

Equivalent construction through SU(5)

Start with the block-diagonal Standard Model embedding in $SU(5)$ , pass to complex Lie algebras

\mathfrak g_{\mathrm{SM}}\subset\mathfrak{sl}_5(\mathbb C),

and embed $\mathfrak{sl}_5$ as a regular $A_4$ -subalgebra of $\mathfrak e_7$ . All $A_4$ root subsystems of $E_7$ lie in one Weyl-group orbit, so this gives the same automorphism class.

Why the qualification matters

An abstract inclusion of a copy of $\mathfrak{sl}_3\oplus\mathfrak{sl}_2\oplus\mathbb C$ in $\mathfrak e_7$ need not have the centralizers and branching rules used in the three-generation construction. The adjective “good” records the regular-position hypothesis from which the generation $\mathfrak{sl}_3$ , standard $\mathfrak{sl}_6$ , and subsequent decompositions follow.

Dependence on choices

The definition is invariant under automorphisms of $\mathfrak e_7$ . Choosing a particular representative embedding is still necessary to regard the constructed algebras as literal subalgebras rather than only as conjugacy classes. Later choices of Cartan subalgebra and roots are additional and are not part of “goodness.”

References

John C. Baez, “Three Generations in E7,” 2026, §2. arXiv:2608.06271.
Toshio Oshima, “A Classification of Subsystems of a Root System,” 2006, especially the tables of subsystem orbits. arXiv:math/0611904.
E. B. Dynkin, “Semisimple Subalgebras of Semisimple Lie Algebras,” American Mathematical Society Translations, Series 2, 6 (1957), 111–244.