Let

$$\pi(x, y, z)=\frac{x+i y}{1-z}$$

be stereographic projection from the unit sphere $S^{2}$ in $\mathbb{R}^{3}$ to the Riemann sphere $\mathbb{C}_{\infty}$. Show that if $r$ is a rotation of $S^{2}$, then $\pi r \pi^{-1}$ is a Möbius transformation of $\mathbb{C}_{\infty}$ which can be represented by an element of $S U(2)$. (You may assume without proof any result about generation of $S O(3)$ by a particular set of rotations, but should state it carefully.)