It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$.
My question is ¿are there more embeddings $U(2) \hookrightarrow SO(4)$ or are they completely parametrized by $S^2$?