This question seems fairly trivial but I'm struggling to find an elegant way to determine such. Suppose that you have a unitary matrix $U$ (not necessarily a density matrix) and you want to know if $U=U_1\otimes U_2$, for a given $U_1$ and $U_2$. This question is not trivial in the sense that if it was easy to determine such for any dimension of $U,U_1$ and $U_2$, then it would be easy to know if a density matrix is separable, nevertheless it remains as an NP-hard problem.
However, for the case where $U$ is a two qubit gate it should be relatively simple. Is there a criteria I could use to show if $U$ (4x4) is is separable into two single-qubit gates?
The only thing I could get is to diagonalize $U$ and check if there are some common factors.