Exercise 3, § 5

Let $\mathfrak{h}$ be a splitting Cartan subalgebra of $\mathfrak{g},\mathfrak{b}$ a Borel subalgebra of $(\mathfrak{g},\mathfrak{h})$, $\mathfrak{n}= [\mathfrak{b},\mathfrak{b}]$ and N = exp ad$_{\mathfrak{g}}\mathfrak{n}$. Then

Aut($\mathfrak{g}$) $= N$.Aut($\mathfrak{g},\mathfrak{h}$)$.N$.

(Let $s\in$ Aut($\mathfrak{g}$). Apply Prop. 10 of §3, no. 3 to $\mathfrak{b}\cap s(\mathfrak{b})$, then apply Chap. VII, §3, no. 4, Th. 3.)