Chevalley Basis

Let $G$ be split semisimple with Lie algebra $\mathfrak g$ and split Cartan $\mathfrak t$.

A Chevalley basis is a basis of the form $\{H_i\}\cup\{X_\alpha\}_{\alpha\in\Phi}$ where:

• each $X_\alpha$ spans the root space $\mathfrak g_\alpha$,
• the brackets satisfy $[X_\alpha,X_{-\alpha}]=H_\alpha$,
• all structure constants are integers.

In the letter: choosing root vectors $X_\alpha$ “satisfying Chevalley conditions” rigidifies automorphisms (the group $\Omega$) and supports integral forms.