Corollary 2

Let $w\in W$. There exists an elementary automorphism $\varphi$ of $\mathfrak{g}$ that commutes with $\theta$, leaves $\mathscr{G}$ and $\mathscr{U}$ stable, and extends $w$.

It suffices to treat the case in which $w$ is of the form $s_{\alpha}(\alpha \in B)$. Note first of all that ad $x_{\alpha}$ and ad $y_{\alpha}$ are locally nilpotent on $U(\mathfrak{g})$, in other words that for all $u\in U(\mathfrak{g})$ there exists an integer $n$ such that (ad $x_{\alpha}$)$^nu=$ (ad $y_{\alpha}$)$^nu= 0$.

This enables us to define the automorphisms $e^{adx_{\alpha}}=\sum_{n=0}^{\infty}\frac{1}{n!}$(ad $x_{\alpha}$)$^n$ and $e^{ady_{\alpha}}$ of $U(\mathfrak{g})$; we verify immediately that these automorphisms of $U(\mathfrak{g})$ leave $\mathscr{U}$ stable. Put $\varphi_1=e^{adx_{\alpha}}e^{ady_{\alpha}}e^{adx_{\alpha}},\varphi_2=e^{ady_{\alpha}}e^{adx_{\alpha}}e^{ady_{\alpha}}$. We have $\varphi_1|\mathfrak{g}=$ $\varphi_2|\mathfrak{g}($§2, no. 2, formula (1)), so $\varphi_1=\varphi_2$. Put $\varphi_1=\varphi_2=\varphi$. We have $\theta \varphi \theta^{-1}=\varphi$, so $\theta$ and $\varphi$ commute. On the other hand, $\varphi |\mathfrak{h}=w$ by §2, no. 2, Lemma 1.