Q:

Does $SU(2)\cong Sp(1)\subset SO(5)$ factor through $Spin(5)\cong Sp(2)$ as the standard embedding $Sp(1) \to Sp(2)$?

$SU(2)$ can be seen as a subgroup of $SO(5)$ through the following chain of subgroups
$$
SU(2) \subset SO(4) \subset SO(5).
$$
If we identify $SU(2)\cong Sp(1)$, does the inclusion $Sp(1) \to SO(5)$ factor through $Spin(5) \cong Sp(2) \to SO(5)$ as the standard embedding of $Sp(1)$ in $Sp(2)$. I understand, that there is a lift of $Sp(1) \to SO(5)$ to $Sp(1) \to Sp(2)$, but I can not see that this is the standard embedding.
I already asked this also at mathstackex.

A:

Yes.  The rep of $Sp(2)$ on $\mathbb{C}^5$ that induces the covering map is the complement to the line spanned by the symplectic form in $\bigwedge{}^{2}\mathbb{C}^4$.  What you are asking is whether there is another line in this space which is invariant under $Sp(1)$, that is, if the action of $Sp(1)$ on the wedge square $\bigwedge{}^{2}\mathbb{C}^4$ fixes a 2-d subspace.  Since the representation on $ \mathbb{C}^4$ of $Sp(1)\cong SU(2)$ is the defining rep on $\mathbb{C}^2$ and two trivials, you get two invariant 2-forms from the symplectic form on $\mathbb{C}^2$ and wedging the two trivials.

