Exercise 4, § 3

Assume that $k=\mathbf{R}$ or $\mathbf{C}$. Put G = Int($\mathfrak{g}$), and identify the Lie algebra of G with $\mathfrak{g}$. Let $\mathfrak{b}$ be a Borel subalgebra of $(\mathfrak{g},\mathfrak{h})$, and $\mathfrak{n}$ the set of its nilpotent elements. Denote by $H,B,N$ the integral subgroups of G with Lie algebras $\mathfrak{h},\mathfrak{b},\mathfrak{n}$, respectively. Show that $H,B,N$ are Lie subgroups of G, that N is simply-connected, and that B is the semi-direct product of H by N.