Exercise 6, § 2

Let $(X_{\alpha})_{\alpha\in R}$ be a Chevalley system in the split semi-simple Lie algebra $(\mathfrak{g},\mathfrak{h})$. Let $\alpha , \beta \in R$, and let $p$ (resp. $q$) be the largest integer $j$ such that $\beta +j\alpha \in R$ (resp. $\beta -j\alpha \in R$), cf. Lemma 4. Show that

ad($X_{\alpha}$)$^k(X_{\beta-q\alpha}) =\pm k$!$X_{\beta+(k-q)\alpha}$ for $0\leq k\leq p+q$.

Deduce that the algebra $\mathfrak{g}_{\mathbf{Z}}$ of Prop. 8 is stable under the ad($X_{\alpha}$)$^k/k$! and under the $e^{ad(X_{\alpha})}$ (cf. §12).