There are isomorphisms of topological groups:
1. $\operatorname{SO}(3) \cong \operatorname{SU}(2)/\{\pm I\} =: \mathrm{PSU}(2)$.
2. $\operatorname{SO}(4) \cong (\operatorname{SU}(2) \times \operatorname{SU}(2))/\{\pm(I,I)\}$.
3. $\operatorname{U}(2) \cong (\operatorname{U}(1) \times \operatorname{SU}(2))/\{\pm(I,I)\}$.