Jacobi Identity

Problem

Prove Jacobi’s identity $[A, [B, C]] + [B, [C, A]] + [C, [A,B]] = 0$.

Remember that $[A, B] = AB - BA$ and the operators cannot be assumed to be commutative.

Solution

$[A,[B,C]]=[A,BC-CB]$

$[A,[B,C]]=A(BC-CB)-(BC-CB)A$

$[A,[B,C]]=ABC-ACB-BCA+CBA$

Repeating this process for the other two brackets will lead to:

$[B,[C,A]]=BCA-BAC-CAB+ACB$

$[C,[A,B]]=CAB-CBA-ABC+BAC$.

Note that the order of the operators matter. Nevertheless, every product has a negative pair, so

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