Let $A(X,Y)$ be a matrix with coefficients in $\mathbb{C}[X,X^{-1},Y,Y^{-1}]$, where the conjugation operation is defined on this ring by ${(z X^a Y^b)}^* =\bar{z}X^{-a}Y^{-b}$. So $A$ is said to be unitary, or *paraunitary* as I've seen it, if $^t\bar{A}(X^{-1},Y^{-1})A(X,Y)=\mathbb{I}$.

Is it always possible to write $A$ as the product of diagonal paraunitary matrices and ordinary unitary matrices on $\mathbb{C}$? This happens to be true in the univariate case, with coefficients in $\mathbb{C}[X,X^{-1}]$, and somehow I can't really imagine the generalization is that hard to decide, even if I'm stuck on it. Surprisingly, even in the univariate case, I can't find a clearer reference than this one, which is hardly the mainstream algebra book I would have expected. A important keyword seems to be "McMillan", but that's not really enough to find relevant articles.

I should probably mention that one point is actually to prove that if two paraunitary matrices have the same determinant, then there exists a continuous path of paraunitary matrices joining them, which would be an immediate consequence of the question I'm asking.