Exercise 4, § 3

(a) Let q be a quadratic form of signature (1, 2) on $\mathbf{R}^3$. Let P be the set of $m \in \mathbf{R}^3$ such that $q(m) > 0$. Then P is the union of two disjoint convex cones C and $-C$. If $s \in \mathbf{SO}(q)$, then either $s(C) = C$ and $s(-C) = -C$ or $s(C) = -C$ and $s(-C) = C$. Let $\mathbf{SO}^+(q)$ be the set of $s \in \mathbf{SO}(q)$ such that $s(C) = C$. Then $\mathbf{SO}^+(q)$ is an open normal subgroup of $\mathbf{SO}(q)$ of index 2 in $\mathbf{SO}(q)$. (b) By imitating the method of Exercise 3, define a morphism of $\mathbf{SL}(2, \mathbf{R})$ onto $\mathbf{SO}^+(q)$ with kernel $\{I, -I\}$. Deduce that the real Lie group $\mathbf{SO}^+(q)$ is isomorphic to the real Lie group $\mathbf{SL}(2, \mathbf{R})/\{I, -I\}$.

(c) For $(\xi, \eta) \in \mathbf{C}^2$ and $(\xi', \eta') \in \mathbf{C}^2$, let $f((\xi, \eta), (\xi', \eta')) = \xi \overline{\xi'} - \eta \overline{\eta'}$. Show that the inner automorphism of $\mathbf{SL}(2, \mathbf{C})$ defined by $\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -i \\ -i & 1 \end{pmatrix}$ maps $\mathbf{SU}(f)$ to $\mathbf{SL}(2, \mathbf{R})$.