Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. $\omega(\cdot, J\cdot)$ gives the usual dot product).

Suppose we have $n$ 2-planes $P_i$ which give an $\omega$-orthogonal decomposition $\mathbb{R}^{2n}=\oplus_{i=1 \ldots n} P_i$ (ie. $\omega(P_i, P_j)=0$ for $i \neq j$). Then both $SO(P_1) \times \ldots \times SO(P_n)$ and $SL(P_1) \times \ldots \times SL(P_n)$ acts as symmetries summandwise.

We also have two symmetry groups $SO(2n), Sp_{2n}$ acting on $\mathbb{R}^{2n}$ defined relative to the fixed structure $(\omega, J)$. The question is then: determine the intersections $SO(2n) \cap. SO(P_1) \times \ldots \times SO(P_n)$ and $Sp_{2n} \cap. SL(P_1) \times \ldots \times SL(P_n)$.