Commutators of small rank and reducibility of operator semigroups

It is easy to see that if $\text{cG}$ is a non-abelian group of unitary matrices, then for no members $A$ and $B$ of $\text{cG}$ can the rank of $AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup $\text{cS}$ of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, $\text{cS}$ is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum ${\text{cG}}_1\oplus {\text{cG}}_2$ where ${\text{cG}}_1$ is at most $3\times 3$ and ${\text{cG}}_2$ is abelian.