Signed angle between two vectors without a reference plane

Question

(In three dimensions) I\'m looking for a way to compute the signed angle between two vectors, given no information other than those vectors. As answered in this question, it is

Answer 1

If all you want is a consistent result, then any arbitrary way of choosing between a × b and b × a for your normal will do. Perhaps pick the one that's lexicographically smaller?

(But you might want to explain what problem you are actually trying to solve: maybe there's a solution that doesn't involve computing a consistent signed angle between arbitrary 3-vectors.)