Generation plane in e7

Definition

Fix a good $\mathfrak g_{\mathrm{SM}}\subset\mathfrak e_7$ , its generation algebra $\mathfrak{sl}_3^{\mathrm{gen}}$ , and compatible Cartan subalgebras. If $A$ is the resulting $A_2$ root system of $\mathfrak{sl}_3^{\mathrm{gen}}$ inside the real span $V$ of the $E_7$ roots, the generation plane is

P:=\operatorname{span}_{\mathbb R}(A)\subset V.

It is a two-dimensional Euclidean subspace. Its complexification $\mathbb C\otimes_{\mathbb R}P$ is the Cartan subalgebra of $\mathfrak{sl}_3^{\mathrm{gen}}$ .

Geometry of the A2 roots

Let $w_1,w_2,w_3\in P$ be the weights of the defining $\mathfrak{sl}_3$ -module, normalized so that the roots have squared length $2$ . Then

w_1+w_2+w_3=0, \qquad \lVert w_i\rVert^2=\frac23, \qquad \langle w_i,w_j\rangle=-\frac13\quad(i\ne j),

and

A=\{w_i-w_j:i\ne j\}.

The six roots form three unoriented lines in $P$ , one for each generation $\mathfrak{sl}_2$ .

What is canonical

The subalgebra $\mathfrak{sl}_3^{\mathrm{gen}}$ is determined by the embedded $\mathfrak g_{\mathrm{SM}}$ , but $P$ as a plane in a chosen real root space uses a compatible Cartan subalgebra and the Killing-form identification of Cartan and dual Cartan. Permuting the weights relabels the three generations; replacing a root by its negative changes an orientation convention but not its root line.

Role in decompositions

Orthogonal projection $\pi:V\to P$ partitions the $E_7$ roots through the root-projection trichotomy. The Cartan summand $\mathbb C\otimes P$ , rather than all of $\mathfrak{sl}_3^{\mathrm{gen}}$ , appears in the three-generation vector-space decomposition.

References

John C. Baez, “Three Generations in E7,” 2026, §§3–5. arXiv:2608.06271.