Exercise 7, § 3

(a) Let G be the set of quaternions of norm 1. This is a Lie subgroup of the real Lie group $\mathbf{H}^*$, homeomorphic to $\mathbf{S}_3$ and hence simply connected, cf. General Topology, Chapter XI.

(b) Let E be the real vector space of pure quaternions, identified with $\mathbf{R}^3$ by

For all $g \in G$, we define an automorphism $\varphi(g)$ of E by $\varphi(g)q = gqg^{-1}$. Show that $\varphi$ is a morphism of the Lie group G onto the real Lie group $\mathbf{SO}(3, \mathbf{R})$ with kernel $\{1, -1\}$. Deduce that the Lie group $\mathbf{SO}(3, \mathbf{R})$ is isomorphic to the Lie group $G/\{1, -1\}$.

(c) Let $\mathbf{C}$ be identified with the subfield $\mathbf{R} + \mathbf{R}i$ of $\mathbf{H}$. The mapping $(\lambda, q) \mapsto q\lambda$ of $\mathbf{C} \times \mathbf{H}$ into $\mathbf{H}$ makes $\mathbf{H}$ into a vector space over $\mathbf{C}$ which is identified with $\mathbf{C}^2$ by the choice of basis $(1, j)$. For all $g \in G$, let $\sigma(g)$ be the automorphism of this vector space defined by $\sigma(g)q = gq$. Show that $\sigma$ is an isomorphism of the real Lie group $G$ onto the real Lie group $\mathbf{SU}(2, \mathbf{C})$. The latter is therefore simply connected by (a).

(d) For $(q_1, q_2) \in G \times G$, let $\tau(q_1, q_2)$ be the mapping $q \mapsto q_1qq_2$ of $\mathbf{H} = \mathbf{R}^4$ into itself. Show that $\tau$ is a morphism of the real Lie group $G \times G$ onto the real Lie group $\mathbf{SO}(4, \mathbf{R})$ with kernel $N = \{(1, 1), (-1, -1)\}$ and hence that $\mathbf{SO}(4, \mathbf{R})$ is isomorphic to $(G \times G)/N$.