Matrix multiplication distributes over commutation

$[A,BC]=[A,B]C+B[A,C]$

Proof

$$\begin{array}{rll}[A,BC]& =ABC-BCA& \text{Adding "zero"}\\ & =ABC-BAC+BAC-BCA& \text{Rearranging the expression}\\ & =(AB-BA)C+B(AC-CA)& \text{Applying the definition of commutator}\\ & =[A,B]C+B[A,C]& \end{array}$$

where $[\cdot ,\cdot ]$ is the Commutator.