Given any commutative $*$-ring $R$ of uneven characteristic, is it true that for every square matrix $M$ and unitary matrix $W$, if $W^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W$ is equal to $\left[\begin{matrix}\Sigma & 0\\0 & - \Sigma\end{matrix}\right]$ where $\Sigma$ is Hermitian, then there is a unitary matrix $U$ and unitary matrix $V$ such that $(W')^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W' = 2\left[\begin{matrix}\Sigma & 0\\0 & - \Sigma\end{matrix}\right]$ where $W' = \begin{bmatrix}U & -U \\ V & V \end{bmatrix}$?
I'm interested in proving it for the $*$-ring $\mathbb C \oplus \mathbb C$ where the involution is $(a,b) \mapsto (b,a)$. But a more general result would be great.
