Remark 4

By Prop. 5 of §4, no. 4, there exists a unique automorphism $\theta$ of $\mathfrak{g}$ such that $\theta (x_{\alpha}) =y_{\alpha}$ and $\theta (y_{\alpha}) =x_{\alpha}$ for all $\alpha \in B$, and $\theta (h) =-h$ for all $h\in \mathfrak{h}$; we have $\theta^2= 1$. By construction of $\mathscr{U}$, we see that the automorphism of $U(\mathfrak{g})$ that extends $\theta$ leaves $\mathscr{U}$ stable.