Hall-Witt identity

Use the group commutator convention $[x,y]=x^{-1}{y}^{-1}xy$ and write $u^v=v^{-1}uv$ for conjugation. The Hall-Witt identity isIt holds in every group. Substituting the definitions and cancelling adjacent inverse letters proves the formula directly; the three cyclic factors are all necessary for that cancellation.

Modulo the fourth term of the lower central series, the triple group commutators are central, so the outer conjugations disappear. Replacing an inverse input changes the corresponding leading group commutator by its inverse. In additive notation for the graded Lie algebra associated with the lower central series, the identity becomesthe Jacobi identity. This explains how a group-theoretic cancellation identity supplies the Lie-theoretic compatibility of leading commutators.