Intersection of the three A1 + D6 subalgebras in e7

Statement

For the three generation root lines, set

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

Then

\mathfrak m_1\cap\mathfrak m_2\cap\mathfrak m_3 =\mathfrak{sl}_6^{\mathrm{SM}} \oplus(\mathbb C\otimes_{\mathbb R}P),

where $P$ is the generation plane. In fact, for every $i\ne k$ ,

\mathfrak m_i\cap\mathfrak m_k =\mathfrak m_1\cap\mathfrak m_2\cap\mathfrak m_3.

These are equalities of embedded Lie subalgebras, not merely vector-space isomorphisms. On the right, the standard $\mathfrak{sl}_6$ commutes with the two-dimensional abelian Cartan algebra $\mathbb C\otimes P$ , so the displayed sum is also a Lie-algebra direct sum.

Root-space calculation

With $\Phi_0$ and $\Phi_k$ from the generation-plane partition,

\mathfrak m_k =\mathfrak h\oplus \bigoplus_{r\in\{\pm\beta_k\}\sqcup\Phi_0\sqcup\Phi_k} (\mathfrak e_7)_r.

For $i\ne k$ , the only root spaces common to $\mathfrak m_i$ and $\mathfrak m_k$ are those indexed by $\Phi_0$ . The full Cartan $\mathfrak h$ remains and splits as the Cartan of $\mathfrak{sl}_6^{\mathrm{SM}}$ plus $\mathbb C\otimes P$ .

Dependence on choices

The standard $\mathfrak{sl}_6$ is intrinsic to the good Standard Model embedding, but the three $\mathfrak m_k$ and the explicit Cartan factor $\mathbb C\otimes P$ require a compatible Cartan choice. Relabeling the root lines permutes the $\mathfrak m_k$ and leaves their common intersection unchanged.

References

John C. Baez, “Three Generations in E7,” 2026, Proposition 7. arXiv:2608.06271.