Exercise 4, § 10

Assume that $k=\mathbf{C}$. Let G = Int($\mathfrak{g}$) $=$ Aut$_0(\mathfrak{g})$, and let B be an integral subgroup of G whose Lie algebra $\mathfrak{b}$ is a Borel subalgebra of $\mathfrak{g}$. Show that B is the normalizer of $\mathfrak{b}$ in G (use Exerc. 4 of §3 and Exerc. 11 of §5). Deduce by means of Exerc. 3 that $G/B$ is compact.