Suppose we have an orthogonal/unitary matrix $T$ of even dimension. Then we can decompose it into: $$ T = \begin{bmatrix} U_1 & 0 \\ 0 & U_2 \end{bmatrix} \begin{bmatrix} R & -\sqrt{I-R^2} \\ \sqrt{I-R^2} & R \end{bmatrix} \begin{bmatrix} V_1 & 0 \\ 0 & V_2 \end{bmatrix} $$ where $U_1$, $U_2$, $V_1$, and $V_2$ are orthogonal/unitary matrices half the size of $T$, and $R$ is diagonal.
Does this decomposition have a name? Anything useful known about it?