Editing Commutator (section)

==== Exponential identities ====
Consider a ring or algebra in which the [[exponential function|exponential]] <math>e^A = \exp(A) = 1 + A + \tfrac{1}{2!}A^2 + \cdots</math> can be meaningfully defined, such as a [[Banach algebra]] or a ring of [[formal power series]].

In such a ring, [[Hadamard's lemma]] applied to nested commutators gives: <math display="inline">e^A Be^{-A}
  \ =\  B + [A, B] + \frac{1}{2!}[A, [A, B]] + \frac{1}{3!}[A, [A, [A, B]]] + \cdots
  \ =\  e^{\operatorname{ad}_A}(B).
</math> (For the last expression, see ''Adjoint derivation'' below.) This formula underlies the [[Baker–Campbell–Hausdorff formula#An important lemma|Baker–Campbell–Hausdorff expansion]] of log(exp(''A'') exp(''B'')).

A similar expansion expresses the group commutator of expressions <math>e^A</math> (analogous to elements of a [[Lie group]]) in terms of a series of nested commutators (Lie brackets),
<math display="block">e^A e^B e^{-A} e^{-B} =
\exp\!\left(  [A, B] + \frac{1}{2!}[A{+}B, [A, B]] + \frac{1}{3!} \left(\frac{1}{2} [A, [B, [B, A]]] + [A{+}B, [A{+}B, [A, B]]]\right) + \cdots\right). </math>