Commutators are bilinear

The Commutator satisfies $[xA+yB,C]=x[A,C]+y[B,C]$

Proof

$$\begin{array}{rll}[xA+yB,C]& =(xA+yB)C-C(xA+yB)& \text{Expanding out the multiplications}\\ & =xAC+yBC-xCA-yCB& \text{Rearranging the algebraic expression}\\ & =x(AC-CA)+y(BC-CB)& \text{Applying the definition of Commutator}\\ & =x[A,C]+y[B,C]& \end{array}$$