Lemma 1

Let V be a $\mathfrak{g}$-module and $\rho$ the corresponding representation of $\mathfrak{g}$.

(i) If $a$ is a nilpotent element of $\mathfrak{g}$, and if $\rho (a)$ is locally nilpotent,

for all $b\in \mathfrak{g}$.

(ii) If $\alpha \in R$ and if the images under $\rho$ of the elements of $\mathfrak{g}^{\alpha}$ and $\mathfrak{g}^{-\alpha}$ are locally nilpotent, the set of weights of V is stable under the reflection $s_{\alpha}$.

With the assumptions in (i), we have $\rho$((ad $a$)$^nb$) $=$ (ad $\rho (a)$)$^n\rho (b)$ for all $n\geq 0$, so $\rho (e^{ada}b) =e^{ad\rho(a)}\rho (b)$. On the other hand,

is assertion (ii) of Chap. VII, §3, no. 1, Lemma 1.

We now adopt the assumptions in (ii). Let $\theta_{\alpha}=e^{adX_{\alpha}}e^{adX_{-\alpha}}e^{adX_{\alpha}}$. By (i), there exists $S\in \mathbf{G}\mathbf{L}(V)$ such that $\rho (\theta_{\alpha}b) = S\rho (b)S^{-1}$ for all $b\in \mathfrak{g}$. Now $\theta_{\alpha}|\mathfrak{h}$ is the transpose of $s_{\alpha}($§2, no. 2, Lemma 1). Let $\lambda$ be a weight of V. There exists a non-zero element $x$ of V such that $\rho (h)x=\lambda (h)x$ for all $h\in \mathfrak{h}$. Then

for all $h\in \mathfrak{h}$. Consequently, $s_{\alpha}\lambda$ is a weight of V.