Show that $\alpha$ is continuous and that the image of $\alpha$ is contained in $\mathrm{SO}(\mathbb{H})$. One may begin by showing that $\alpha(u,v) \in \mathrm{O}(\mathbb{H})$ for $(u,v) \in S \times S$.
Show that $\alpha$ is continuous and that the image of $\alpha$ is contained in $\mathrm{SO}(\mathbb{H})$. One may begin by showing that $\alpha(u,v) \in \mathrm{O}(\mathbb{H})$ for $(u,v) \in S \times S$.