Commutator derivation identity
= Commutator derivation identity
{title2=$[A,BC]=[A,B]C+B[A,C]$}
Expand both sides using the definition of a <commutator>: the middle terms $BAC$ cancel, leaving $ABC-BCA$. Thus commutation with a fixed operator is a <derivation of an algebra>. In particular, $[A,[B,C]]=[[A,B],C]+[B,[A,C]]$, which gives a short way to derive a <Lie algebra> representation from generator actions on an underlying algebra.