Proposition 1

Let $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B})$ be a framed semi-simple Lie algebra, and R the root system of $(\mathfrak{g},\mathfrak{h})$. Let G be the set of $s\in$ Aut($\mathfrak{g},\mathfrak{h}$) that leave B stable, and such that $s(X_{\alpha}) =X_{\varepsilon(s)\alpha}$ for all $\alpha \in B ($in other words the set of automorphisms of $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B}))$. Then the restriction of $\varepsilon$ to G is an isomorphism from G to Aut(R$,B$).

If $s\in G$, it is clear that $\varepsilon (s)\in$ Aut(R$,B$). On the other hand, the map

$\varepsilon |G : G\rightarrow$ Aut(R$,B$)

is bijective by Th. 2 of §4, no. 4.