Proposition 2

The multiplicative group $\mathbf{H}^*$ of non-zero quaternions is a topological group isomorphic to the product of its subgroups $\mathbf{R}_+^*$ and $S_3$.

Every quaternion $x / o$ can be written as $x \cdot z$ where $z$ is a quaternion with norm 1; since $\|xx'\| = \|x\|\cdot\|x'\|$, the mapping $x \to (\|x\|, x_1, x_2)$ of $H^*$ onto $\mathbf{R}_+^* \times S_2$ is an isomorphism of the group structures, and from Chapter VI, § 2, no. 2, Proposition 2, it is a homomorphism of $H^*$ onto $\mathbf{R}_+^* \times S_2$.