Corollary

If G is simply-connected, or if C(G) reduces to the identity element, the group Aut(G)$/$Int(G) is isomorphic to the group of automorphisms of the Dynkin graph of $R(G,T)$.

This follows from the preceding and Chap. VI, §4, no. 2, Cor. of Prop. 1.

We are now going to show that the extension (16) admits sections.

For all $\alpha \in R(G,T)$, denote by $V(\alpha )$ the 2-dimensional vector subspace of $\mathfrak{g}$ such that $V(\alpha )_{(\mathbf{C})}=\mathfrak{g}^{\alpha}+\mathfrak{g}^{-\alpha}$; denote by K the quadratic form associated to the Killing form of $\mathfrak{g}$.