Proposition 5

Let $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B})$ be a framed semi-simple Lie algebra, and $(X_{\alpha})_{\alpha\in B\cup(-B)}$ the corresponding generating family. There exists a unique automorphism $\theta$ of $\mathfrak{g}$ such that $\theta (X_{\alpha}) =X_{-\alpha}$ for all $\alpha \in B\cup (-B)$. We have $\theta^2=$ Id$_{\mathfrak{g}}$, and $\theta (h) =-h$ for all $h\in \mathfrak{h}$.

The uniqueness is clear since $(X_{\alpha})_{\alpha\in B\cup(-B)}$ generates the Lie algebra $\mathfrak{g}$. In view of Prop. 4, the existence of $\theta$ follows from what we said in no. 3 before Th. 1.