a) Show that the restriction map to $\mathbb{H}^{\mathrm{im}}$ induces a group isomorphism $$\mathrm{Aut}(\mathbb{H}) \simeq \mathrm{SO}(\mathbb{H}^{\mathrm{im}}).$$ b) Show that $$\mathrm{Aut}(\mathbb{H}) = \{\alpha(u,u) \mid u \in S\}.$$
a) Show that the restriction map to $\mathbb{H}^{\mathrm{im}}$ induces a group isomorphism
$$\mathrm{Aut}(\mathbb{H}) \simeq \mathrm{SO}(\mathbb{H}^{\mathrm{im}}).$$
b) Show that
$$\mathrm{Aut}(\mathbb{H}) = \{\alpha(u,u) \mid u \in S\}.$$