Lemma 9

Let $\mu, \nu \in Q$ and $w\in W(R)$ be such that $w\mu=\nu$. There exists an automorphism of $\mathfrak{g}$ that takes $\mathfrak{g}^\mu$ to $\mathfrak{g}^{\nu}$.

For all $\alpha \in B$, let $s_{\alpha}$ be the reflection in V defined by $\alpha$. Since W(R) is generated by the $s_{\alpha}$ (Chap. VI, §1, no. 5, Remark 1), it suffices to prove the lemma when $w=s_{\alpha}$. In view of Lemma 8, we can define

It is verified as in Chap. I, §6, no. 8, that $\theta_{\alpha}$ is an automorphism of $\mathfrak{g}$. We have

$^1$ An endomorphism $u$ of a vector space V is called locally nilpotent (or almost

nilpotent) if, for every $v\in V$, there exists a positive integer $n$ such that $u^n(v) = 0$

(cf. Commutative Algebra, Chap. IV, §1, no. 4, Def. 2). Then exp($u$), or $e^u$, is

defined by the formula $e^u(v) =\sum_{n\geq 0}(1/n$!)$u^n(v)$ for all $v\in V$. $\theta_{\alpha}(H_{\alpha}) =e^{adX_{\alpha}}e^{adX_{-\alpha}}(H_{\beta}-n(\alpha , \beta )X_{\alpha})$

If $z\in \mathfrak{g}^\mu$,

so $\theta_{\alpha}^{-1}z\in \mathfrak{g}^{s_{\alpha}\mu}$. This shows that $\theta^{-1}_{\alpha}\mathfrak{g}^\mu\subset \mathfrak{g}^{s_{\alpha}\mu}$. Since $\theta_{\alpha}$ is an automorphism and since this inclusion holds for all $\mu\in Q$, we see that $\theta_{\alpha}^{-1}\mathfrak{g}^\mu=\mathfrak{g}^{s_{\alpha}\mu}$, which proves the lemma.