Exercise 6, § 3

Let F be the vector space of Hermitian complex matrices with 2 rows and 2 columns. For all $s \in \mathbf{SL}(2, \mathbf{C})$, we define an automorphism $\sigma(s)$ of F by $\sigma(s)(m) = sms^*$. We identify the element $\begin{pmatrix} t + x & y + iz \\ y - iz & t - x \end{pmatrix}$ of F with the element $(t, x, y, z)$ of $\mathbf{R}^4$ and write

Let C be the set of $(t, x, y, z) \in \mathbf{R}^4$ such that $t^2 - x^2 - y^2 - z^2 > 0,\ t > 0$. Let $\mathbf{SO}^+(q)$ be the set of $g \in \mathbf{SO}(q)$ such that $g(C) = C$; it is an open normal subgroup of $\mathbf{SO}(q)$ of index 2 in $\mathbf{SO}(q)$ (cf. Exercise 4). Show that $\sigma$ is a morphism of the underlying real Lie group of $\mathbf{SL}(2, \mathbf{C})$ onto the real Lie group $\mathbf{SO}^+(q)$ with kernel $\{I, -I\}$. (For the surjectivity, use Exercise 5.) Deduce that the real Lie group $\mathbf{SO}^+(q)$ is isomorphic to the underlying real Lie group of $\mathbf{SL}(2, \mathbf{C})/\{I, -I\}$, that is (Exercise 3) the underlying real Lie group of $\mathbf{SO}(3, \mathbf{C})$.