E7 branching under A1 + D6

Statement

For any one of the three mutually centralizing pairs

\mathfrak m_k =\mathfrak{sl}_2(\beta_k)\oplus\mathfrak{so}_{12}(\beta_k) \subset\mathfrak e_7,

the adjoint representation of $\mathfrak e_7$ , restricted to $\mathfrak m_k$ , has the branching rule

\mathbf{133} \;\downarrow_{A_1+D_6} = (\mathbf 3,\mathbf 1) \oplus(\mathbf 1,\mathbf{66}) \oplus(\mathbf 2,\mathbf{32}).

Here $\mathbf3$ and $\mathbf2$ are respectively the adjoint and defining $\mathfrak{sl}_2$ -modules, $\mathbf{66}$ is the adjoint $\mathfrak{so}_{12}$ -module, and $\mathbf{32}$ is one half-spin representation of $\mathfrak{so}_{12}$ .

What the direct sum means

Equivalently, as an $\mathfrak m_k$ -module and hence as a vector space,

\mathfrak e_7 =\mathfrak{sl}_2(\beta_k) \oplus\mathfrak{so}_{12}(\beta_k) \oplus(\mathbf2\otimes\mathbf{32}).

The first two summands together form the Lie subalgebra $\mathfrak m_k$ . The $64$ -dimensional last summand is an invariant module, not a Lie subalgebra and not asserted to be closed under the $\mathfrak e_7$ bracket.

Weight-space explanation

The roots not belonging to the $A_1+D_6$ subsystem pair with $\beta_k$ by $\pm1$ . Thus their root spaces form copies of the defining $\mathfrak{sl}_2$ -module. Either $\pm1$ eigenspace has dimension $32$ , with distinct equal-length $D_6$ weights, identifying it as a half-spin module.

Convention for 32

The two chiral half-spin representations of $\mathfrak{so}_{12}$ are exchanged by a Dynkin-diagram automorphism of $D_6$ . Calling the occurring module $\mathbf{32}$ rather than $\mathbf{32}'$ is therefore a convention; the branching rule is invariant after exchanging the two labels.

References

John C. Baez, “Three Generations in E7,” 2026, Proposition 5. arXiv:2608.06271.
R. Slansky, “Group Theory for Unified Model Building,” Physics Reports 79 (1981), 1–128. DOI record90092-2).
E. B. Dynkin, “Semisimple Subalgebras of Semisimple Lie Algebras,” American Mathematical Society Translations, Series 2, 6 (1957), 111–244.