Definition 3

A Chevalley system for $(\mathfrak{g},\mathfrak{h})$ is a family $(X_{\alpha})_{\alpha\in R}$ such that

(i) $X_{\alpha}\in \mathfrak{g}^{\alpha}$ for all $\alpha \in R$;

(ii) $[X_{\alpha}, X_{-\alpha}] =-H_{\alpha}$ for all $\alpha \in R$;

(iii) the linear map from $\mathfrak{g}$ to $\mathfrak{g}$ which is equal to $-1$ on $\mathfrak{h}$ and which takes $X_{\alpha}$ to $X_{-\alpha}$ for all $\alpha \in R$ is an automorphism of $\mathfrak{g}$.

The extension of this definition to the case where $(\mathfrak{g},\mathfrak{h})$ is split reductive is immediate.

We shall show (§4, no. 4, Cor. of Prop. 5) that Chevalley systems for $(\mathfrak{g},\mathfrak{h})$ exist.