Commutators satisfy the Jacobi Identity

$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$, where

• $[\cdot ,\cdot ]$ is the Commutator

Proof

$$\begin{array}{rll}[A,[B,C]]+[B,[C,A]]+[C,[A,B]]& =[A,BC-CB]+[B,CA-AC]+[C,AB-BA]& \text{By the commutator’s definition}\\ & =A(BC-CB)-(BC-CB)A+B(CA-AC)-(CA-AC)B+C(AB-BA)-(AB-BA)C& \text{Expanding the multiplications}\\ & =ABC-ACB-BCA+CBA+BCA-BAC-CAB+ACB+CAB-CBA-ABC+BAC& \text{Cancelling the repeated terms}\\ & =0& \end{array}$$