Must unitary matrices satisfying this property commute?

Question

If A and B are unitary matrices such that A, B, and AB are all conjugate to diag(1,1,-1,-1), must AB=BA? Why or why not?

Answer 1

This is true, and it has nothing to do with unitarity. Let $g\in \operatorname{GL}_4(\mathbb{C})$ be such that $gAg^{-1}=\operatorname{diag}(1,1,-1,-1)$ so that $$ gA^2g^{-1}=(gAg^{-1})^2=\operatorname{diag}(1,1,-1,-1)^2=1 $$ so $A^2=g^{-1}1g=1$. Likewise, $B^2=1$ and $ABAB=(AB)^2=1$. Therefore \begin{align} AB-BA &=(AB-BA)ABAB\\ &=(ABAB-BAAB)AB\\ &=(1-1)AB\\ &=0. \end{align}