Generation sl2 subalgebras

Core idea

Let $A\subset P$ be the $A_2$ roots in the generation plane. Write

A=\{\pm\beta_1,\pm\beta_2,\pm\beta_3\},

one opposite pair for each of the three root lines. The corresponding generation $\mathfrak{sl}_2$ subalgebras are

\mathfrak{sl}_2(\beta_k) =(\mathfrak e_7)_{\beta_k} \oplus\mathbb C h_{\beta_k} \oplus(\mathfrak e_7)_{-\beta_k} \subset\mathfrak{sl}_3^{\mathrm{gen}}, \qquad k=1,2,3.

Each is the root $\mathfrak{sl}_2$ -subalgebra associated with its root line and is regular. Replacing $\beta_k$ by $-\beta_k$ gives the same subalgebra.

Relation to the defining weights

Choose the weights $w_1,w_2,w_3$ of the defining generation $\mathfrak{sl}_3$ -module and set

\beta_k=w_i-w_j

when $(i,j,k)$ is cyclic. Then $w_k\perp\beta_k$ , while the other two weights pair with $\beta_k$ by $\pm1$ . Thus the $k$ -th root line fixes the weight direction labeled $k$ and mixes the other two.

Centralizers in e7

For each $k$ , the roots orthogonal to $\beta_k$ form the corresponding $D_6$ subsystem. Consequently $\mathfrak{sl}_2(\beta_k)$ and a copy $\mathfrak{so}_{12}(\beta_k)$ are mutual centralizers in $\mathfrak e_7$ .

Dependence on choices

The unordered set of three root-line subalgebras is determined after choosing a Cartan subalgebra of $\mathfrak{sl}_3^{\mathrm{gen}}$ . Their labels $1,2,3$ and the signs of the $\beta_k$ are conventions. Without a Cartan choice, the generation $\mathfrak{sl}_3$ is intrinsic but no distinguished triple of these $\mathfrak{sl}_2$ 's is selected.

References

John C. Baez, “Three Generations in E7,” 2026, §4. arXiv:2608.06271.