A commutator is a sequence of the form X.Y.X-1.Y-1 which is useful for affecting just a few pieces on the cube while leaving everything else untouched. For two arbitrary sequences X and Y, the commutator will affect only the parts of the cube that are "caught up" in the intersection of X and Y. If X and Y are completely disjoint and affect completely different parts of the cube, then obviously X.Y.X-1.Y-1 will do exactly nothing. If X and Y overlap, then X.Y.X-1.Y-1 will affect only the parts of the cube where X and Y overlapped. The idea is to make the overlap as small as possible but not zero.