Here is a discussion of $SO_{3}\cong P^3$ in Basic topology,Armstrong,Page 77. I think $\mathbb H$ below is the space of quaternions.
- We think of $S^3$ as the quaternions of norm 1,and note that conjugation in $\mathbb H$ by a nonzero quaternion always induces a rotation of the three-dimensional subspace of pure quaternions.This defines a function $\mathbb H-\{0\}\to SO_3$ which is in fact a homomorphism,onto,and continuous......
Well,I can follow except continuity.I don't think it is obvious at all,since $SO_3$ is a subspace of $\mathbb E^9$ while $\mathbb H$ is 4-dimensional vector space.