Fixing notation: for matrices $A,X$ we let $A[X]$ denote ${}^tXAX$.

Let $P_n$ denote the collection of real $n\times n$ positive definite symmetric matrices.

For $Y\in P_n$ we have the usual Iwasawa decomposition $Y=D[U]$, where $D$ is a diagonal with entries $d_i >0$ and $U$ unipotent.

We consider now the following partial Iwasawa decomposition. For $Y\in P_n$

we have a decomposition $$Y=\begin{pmatrix} v & 0 \\\ 0 & W \end{pmatrix} \begin{bmatrix} 1 & {}^tx \\\ 0 & I_{n-1} \end{bmatrix}$$ where $v>0, W\in P_{n-1}, x\in \mathbb{R}^{n-1}$.

NB. The uniqueness of this decomposition can be found, for instance, in Igusa's "Theta Functions", V.4.12.

Our question is then this: how do we describe those elements of $P_{2n}$ which are symplectic? ie. with respect to the partial Iwasawa decomposition above give an effective characterization of the positive definite symmetric symplectic matrices.

We are aware of the usual partial Iwasawa decomposition of symplectic positive definite symmetric $Z$ in the form $$Z=\begin{pmatrix} W & 0 \\\ 0 & W^{-1} \end{pmatrix} \begin{bmatrix} I_g & X \\\ 0 & I_g \end{bmatrix}$$ where $W, X$ real symmetric and $W$ positive definite.