Exercise 1, § 6

Let D be a set of elements of $\mathbf{SL}(2, \mathbf{C})$ of the form $\begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix}$ where $a > 0$ and $b \in \mathbf{C}$. Show that D is a Lie subgroup of the underlying real Lie group of $\mathbf{SL}(2, \mathbf{C})$ and that the mapping $(u, d) \mapsto ud$ of $\mathbf{SU}(2, \mathbf{C}) \times D$ into $\mathbf{SL}(2, \mathbf{C})$ is an isomorphism of real analytic manifolds. Deduce from this and § 3, Exercise 7 (c) that $\mathbf{SL}(2, \mathbf{C})$ is simply connected.