Exercise 5, § 3

Let E be the real vector space of Hermitian complex matrices with 2 rows and 2 columns of zero trace. For all $s \in \mathbf{SU}(2, \mathbf{C})$, we define an automorphism $\varphi(s)$ of E by $\varphi(s)(m) = sms^*$. We identify the element

of E with the element $(x, y, z)$ of $\mathbf{R}^3$. By imitating the method of Exercise 3, show that $\varphi$ is a morphism of the real Lie group $\mathbf{SU}(2, \mathbf{C})$ onto the real Lie group $\mathbf{SO}(3, \mathbf{R})$ with kernel $\{I, -I\}$ and that $\mathbf{SO}(3, \mathbf{R})$ is isomorphic to $\mathbf{SU}(2, \mathbf{C})/\{I, -I\}$