Exceptional chain to the Standard Model Lie algebra

Core idea

For a compatible choice of Cartan subalgebra and simple roots in $\mathfrak e_7$ , repeated simple-root removal gives a chain of regular semisimple Lie subalgebras

\mathfrak{sl}_3\oplus\mathfrak{sl}_2 \subset \mathfrak{sl}_5 \subset \mathfrak{so}_{10} \subset \mathfrak e_6 \subset \mathfrak e_7.

Equivalently, its root-system types are

A_2+A_1\subset A_4\subset D_5\subset E_6\subset E_7.

Inside the $A_4$ stage, delete the node that leaves $A_2+A_1$ but retain the full $A_4$ Cartan. This produces the maximal Levi subalgebra

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

Every displayed inclusion is a Lie-algebra inclusion. This chain is not a direct-sum decomposition of $\mathfrak e_7$ .

Root-system construction

Given simple roots $\alpha_1,\ldots,\alpha_n$ , retain the roots in the integer span of $\alpha_1,\ldots,\alpha_{n-1}$ and the corresponding smaller Cartan subspace. Iterating the operation along the selected $E_7$ diagram produces the exceptional $E_n$ chain

E_7\supset E_6\supset E_5=D_5\supset E_4=A_4\supset E_3=A_2+A_1.

The last equality is an equality of root-system types and hence of the associated complex semisimple Lie algebras.

Role in the E7 construction

The chain supplies a concrete good embedding of $\mathfrak g_{\mathrm{SM}}$ in $\mathfrak e_7$ . It also exhibits the familiar $\mathfrak{sl}_5$ , $\mathfrak{so}_{10}$ , and $\mathfrak e_6$ enlargements in one regular-subalgebra framework.

Dependence on choices

The literal embedded chain depends on a Cartan subalgebra, a simple-root system, and an ordering of the deleted nodes. Its automorphism class is the invariant relevant here. The abelian $\mathbb C$ in $\mathfrak g_{\mathrm{SM}}$ is a Levi-center direction; it is not an extra simple factor in the semisimple root-system chain.

References

John C. Baez, “Three Generations in E7,” 2026, §2 and Table 2. arXiv:2608.06271.
E. B. Dynkin, “Semisimple Subalgebras of Semisimple Lie Algebras,” American Mathematical Society Translations, Series 2, 6 (1957), 111–244.
John C. Baez and John Huerta, “The Algebra of Grand Unified Theories,” Bulletin of the American Mathematical Society 47 (2010), 483–552. arXiv:0904.1556.