Corollary 3

Let $\alpha \in R$. If $x\in \mathscr{G}\cap \mathfrak{g}^{\alpha}$ and $n\in \mathbf{N}$, we have $x^{(n)}\in \mathscr{U}$, and (ad $x$)$^n/n$! leaves $\mathscr{G}$ and $\mathscr{U}$ stable.

This is clear if $\alpha \in B$, by construction of $\mathscr{U}$ and Cor. 1. In the general case, there exists $w\in W$ such that $w(\alpha )\in B$ (Chap. VI, §1, no. 5, Prop. 15). By Cor. 2, there exists an automorphism $\varphi$ of $\mathfrak{g}$ that leaves $\mathscr{G}$ and $\mathscr{U}$ stable and takes $\mathfrak{g}^{\alpha}$ to $\mathfrak{g}^{w(\alpha)}$, hence the corollary by transport of structure.