It is known that $Sp(1) \cong SU(2)$ and since $SU(2)$ is a subgroup of $SO(4)$ it can be regarded as a subgroup of $SO(5)$ (standard embedding). On the other side we have that $SO(4)$ is a subgroup of $Sp(2)$.
- is the embedding of $Sp(1) \subset SO(4)\subset Sp(2)$ the standard embedding?
- $Sp(2)$ is the double cover of $SO(5)$. Does the embedding of $Sp(1)$ into $SO(5)$ from above factor through this covering?