Exercise 5, § 10

Assume that $\mathfrak{g}$ is splittable. If $\mathfrak{h}$ is a splitting Cartan subalgebra of $\mathfrak{g}$, denote by $E(\mathfrak{h})$ the subgroup of Aut$_e(\mathfrak{g})$ generated by the $e^{adx},x\in \mathfrak{g}^{\alpha}(\mathfrak{h})$, $\alpha \in R(\mathfrak{g},\mathfrak{h})$, cf. Chap. VII, §3, no. 2. a) Let $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$ containing $\mathfrak{h}$, and let $\mathfrak{h}_1$ be a Cartan subalgebra of $\mathfrak{b}$. Show that $\mathfrak{h}_1$ is conjugate to $\mathfrak{h}$ by an element of $E(\mathfrak{h})$. Deduce that $E(\mathfrak{h}) = E(\mathfrak{h}_1)$.

b) Let $\mathfrak{h}'$ be a splitting Cartan subalgebra of $\mathfrak{g}$. Show that $E(\mathfrak{g}) = E(\mathfrak{h}')$. (If $\mathfrak{b}'$ is a Borel subalgebra containing $\mathfrak{h}'$, choose a Cartan subalgebra $\mathfrak{h}_1$ of $\mathfrak{b}\cap \mathfrak{b}'$ and apply $a$) to show that $E(\mathfrak{h}) = E(\mathfrak{h}_1) = E(\mathfrak{h}').)$

c) Let $x$ be a nilpotent element of $\mathfrak{g}$. Show that $e^{adx}\in E(\mathfrak{h})$. (Use $b$) and Cor. 2 of Th. 1 to reduce to the case in which $x\in [\mathfrak{b},\mathfrak{b}].)$ Deduce that $E(\mathfrak{h}) =$ Aut$_e(\mathfrak{g})$.