For $X=\mathbb C^N\oplus \mathbb C^N$ equipped with a real structure $J^2=1$ and symplectic structure $S$ satisfying $$J(z_1,z_2)=(\bar z_2,\bar z_1),~~~~~S(z_1,z_2)=(z_1,-z_2)$$ we see that $X$ has the structure of an $N$-dimensional Kahler vector space, and we can define the class $\mathcal A$ of real-valued quadratic forms on $X$, represented by block matrices as $M(z,w)\equiv z^\dagger Mw$: $$\mathcal A \equiv \{M~|~M(z,w)\in \mathbb R\}$$ Note that we can write $$\mathcal A=\left\{M\in M_{2N}(\mathbb C)~\bigg|~M=\begin{bmatrix}A&B^\dagger\\B&A^T\end{bmatrix},~~~A~~\text{Hermitian}\right\}$$ We now consider a subclass of these quadratic forms which are of the form $$\mathcal B\equiv \left\{M\in M_{2N}(\mathbb C)~\bigg|~M=\begin{bmatrix}A&0\\0&A^T\end{bmatrix},~~~A~~\text{Hermitian}\right\}$$
what is the nicest surjective map $\mathcal A\to \mathcal B$, representable by conjugation by a symplectic matrix, that you can come up with? In other words, we want, for each $M\in \mathcal A$, a symplectic matrix $S_M$ such that $S_MMS_M^*\in \mathcal B$.
Motivation: this would represent a canonical reduction of a quadratic quantum field theory to a non-interacting quantum field theory.