Three-generation decomposition of e7

Statement

Fix a good embedded $\mathfrak g_{\mathrm{SM}}\subset\mathfrak e_7$ , a compatible Cartan subalgebra, and a labeling of the three generation root lines. Then

\mathfrak e_7 =\mathfrak{sl}_6^{\mathrm{SM}} \oplus(\mathbb C\otimes_{\mathbb R}P) \oplus V_1\oplus V_2\oplus V_3,

where each $V_k$ is the 32-dimensional generation module. Every summand is invariant under $\mathfrak{sl}_6^{\mathrm{SM}}$ acting through the $\mathfrak e_7$ bracket, and

V_k\cong\Lambda^{\mathrm{even}}\mathbb C^6 \quad\text{as \(\mathfrak{sl}_6^{\mathrm{SM}}\)-modules}, \qquad V_k\cong\Lambda\mathbb C^5 \quad\text{as \(\mathfrak{sl}_5^{\mathrm{SM}}\)-modules}.

Hence each $V_k$ restricts to one full Standard Model generation, including the two neutrino singlet states.

What the direct sum means

The theorem is a direct-sum decomposition of the underlying vector space and of the $\mathfrak{sl}_6^{\mathrm{SM}}$ -module:

133=35+2+32+32+32.

Only

\mathfrak{sl}_6^{\mathrm{SM}} \oplus(\mathbb C\otimes P)

is asserted to be a Lie subalgebra; it is the common intersection of the three $A_1+D_6$ subalgebras. The $V_k$ are invariant linear subspaces, not Lie subalgebras, and the displayed decomposition is not a direct sum of Lie algebras.

Root-space content

The common Lie subalgebra consists of the full Cartan together with the $30$ root spaces in $\Phi_0$ . Each

V_k=\bigoplus_{r\in\{\pm\beta_k\}\sqcup\Phi_k}(\mathfrak e_7)_r

contains $30$ root spaces projected to $\pm w_k$ and the two generation-root spaces projected to $\pm\beta_k$ . The root partition, together with $A=\bigsqcup_{k=1}^3\{\pm\beta_k\}$ , makes the seven root-index sets used here disjoint and exhaustive.

Dependence on choices

The standard $\mathfrak{sl}_6$ is intrinsic to the chosen good Standard Model embedding, but the splitting of the generation $\mathfrak{sl}_3$ into its two Cartan directions and six root spaces is not. Therefore the individually named $V_1,V_2,V_3$ require a Cartan choice and root-line labeling. Without those choices, the intrinsic statement is instead the $A_2+A_5$ branching rule, in which the generation $\mathfrak{sl}_3$ remains unbroken.

Mathematical scope

This theorem is a representation-theoretic pattern inside the complex Lie algebra $\mathfrak e_7$ . By itself it does not specify a physical theory, spacetime spin representation, dynamics, symmetry breaking, or an explanation of observed particle masses and mixings.

References

John C. Baez, “Three Generations in E7,” 2026, Theorem 13. arXiv:2608.06271.
Benjamin Nasmith, “An Exceptional Combinatorial Sequence and Standard Model Particles,” 2020. arXiv:2012.03933.
T. Kugo and T. Yanagida, “Unification of Families Based on a Coset Space $E_7/(SU(5)\times SU(3)\times U(1))$ ,” Physics Letters B 134 (1984), 313–317. DOI record90007-8).