Proposition 6

Let $\mathfrak{h}$ be a splitting Cartan subalgebra of $\mathfrak{g}$. We have, with the notations in no. 1, Aut$_0(\mathfrak{g}) =$ Aut$_e(\mathfrak{g})$.Ker $\varepsilon =$ Aut$_e(\mathfrak{g}).f(T_Q)$.

By §3, no. 3, Cor. of Prop. 10, Aut$_0(\mathfrak{g}) =$ Aut$_e(\mathfrak{g})$.Aut$_0(\mathfrak{g},\mathfrak{h})$. On the other hand, $\varepsilon$(Aut$_e(\mathfrak{g},\mathfrak{h})$)$\supset W(R)$ by §2, no. 2, Cor. of Th. 2, so Aut$_0(\mathfrak{g},\mathfrak{h}) =$ Aut$_e(\mathfrak{g},\mathfrak{h})$.Ker$\varepsilon$.