Problem
Problem 4.72 Prove that $S U(2) \cong S p(1)$ and apply it to prove that $S U(2)$ is a twofold covering of $\mathrm{SO}(3)$.
Hint Apply that $\operatorname{Sp}(1)$ has centre $\mathbb{Z}_{2} \cong\{ \pm I\}$ and acts by conjugation on $\operatorname{Im} \mathbb{H} \cong \mathbb{R}^{3}$.
One can find the relevant theory, for instance, in Ziller [15].