Operator commutator ($[A,B]=AB-BA$)

The commutator measures failure of two operators to commute, on a common domain where the products are defined. Direct expansion gives $[AB,C]=A[B,C]+[A,C]B$. A common eigenvector of $A,B$ lies in the kernel of $[A,B]$, so an injective commutator excludes all nonzero common eigenvectors. A nonzero commutator need not itself be injective, and therefore does not alone exclude an individual common eigenvector.