Two-out-of-three property for unitary structures
= Two-out-of-three property for unitary structures
On $\mathbb R^{2n}$ let $J_0$ be the standard <complex structure>, let $\omega_0$ be the standard <symplectic form>, and let $g_0(u,v)=\omega_0(u,J_0v)$ be the Euclidean <inner product>. A linear map preserving any two of $J_0$, $\omega_0$, and $g_0$ preserves the third. Equivalently,
$$
GL(n,\mathbb C)\cap Sp(2n,\mathbb R)
=Sp(2n,\mathbb R)\cap O(2n)
=O(2n)\cap GL(n,\mathbb C)
=U(n).
$$