When using matrices, one defines the commutator of two n x n matrices A and B to be AB-BA
A property of the commutator is that it is never the
identity (or a non null
multiple thereof). This is proved easily enough using the
trace operator (noted tr).
Suppose that AB-BA = I
then tr(AB-BA) = tr I. The properties of
trace tell us that tr(AB-BA)=tr(AB)-tr(BA)=0
but tr I = n (since I is the n x n identity matrix),
hence (AB-BA) cannot be the identity.