Define $$U = \begin{pmatrix} a+ib && c+id \\ -c+id && a-ib \end{pmatrix}, \quad \text{det}(U)=1.$$
Let $U'\subset\rm{SU(2)}$ be the subset of $U$ s.t. $a>0$. Let $\phi:U'\rightarrow\mathbb{R}^3, \phi(U)=(b, c, d).$ For any element $U\in\rm{SU(2)},$ let $\xi_{U}(P)=\phi(U^{-1}P).$ Let $V_{U}$ be the domain of $\xi_{U}$.
I am told in class that it's possible to find an atlas that consists of 8 charts to cover $\rm{SU(2)}$. One is $(V_{U},\xi_{U})$. Ireally have not much idea about how to construct this atlas. But I'm wondering if it's helps to consider the 8 cases that $a,b,c,d>0,a,b,c,d<0?$