Looking at this and this question about the Lagrangian Grassmannian, and its linked Wikipedia description as the quotient of $Sp(N)$ by the unitary group $U(n)$, I wondering what is the explicit embedding $U(N) \hookrightarrow Sp(2n)$.

As far I understand, we have (also by Wikipedia) that $Sp(n)$ is the group of quaternionic unitary matrices of order $n$, and so, a subset of $M(2n,\mathbb{C})$. By analogy with the case of the usual Grassmannian, I would embed $U(n)$ in the bottom right hand corner, and put $1$'s on the remaining diagonal entries. However, it is not clear to me that this is actually contained in $Sp(n)$. Moreover, will it make a difference if I instead embed it in the top left hand corner. Because of less Dynkin diagram symmetries in the $C$-series case, I feel this should not be isomorphic.