Editing Commutator (section)

=== Identities (ring theory) ===
The commutator has the following properties:

==== Lie-algebra identities ====
# <math>[A + B, C] = [A, C] + [B, C]</math>
# <math>[A, A] = 0</math>
# <math>[A, B] = -[B, A]</math>
# <math>[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0</math>

Relation (3) is called [[anticommutativity]], while (4) is the [[Jacobi identity]].

==== Additional identities ====
# <math>[A, BC] = [A, B]C + B[A, C]</math>
# <math>[A, BCD] = [A, B]CD + B[A, C]D + BC[A, D]</math>
# <math>[A, BCDE] = [A, B]CDE + B[A, C]DE + BC[A, D]E + BCD[A, E]</math>
# <math>[AB, C] = A[B, C] + [A, C]B</math>
# <math>[ABC, D] = AB[C, D] + A[B, D]C + [A, D]BC</math>
# <math>[ABCD, E] = ABC[D, E] + AB[C, E]D + A[B, E]CD + [A, E]BCD</math>
# <math>[A, B + C] = [A, B] + [A, C]</math>
# <math>[A + B, C + D] = [A, C] + [A, D] + [B, C] + [B, D]</math>
# <math>[AB, CD] = A[B, C]D + [A, C]BD + CA[B, D] + C[A, D]B =A[B, C]D + AC[B,D] + [A,C]DB + C[A, D]B</math>
# <math>[[A, C], [B, D]] = [[[A, B], C], D] + [[[B, C], D], A] + [[[C, D], A], B] + [[[D, A], B], C]</math>

If {{mvar|A}} is a fixed element of a ring ''R'', identity (1) can be interpreted as a [[product rule|Leibniz rule]] for the map <math>\operatorname{ad}_A: R \rightarrow R</math> given by <math>\operatorname{ad}_A(B) = [A, B]</math>.  In other words, the map ad<sub>''A''</sub> defines a [[derivation (abstract algebra)|derivation]] on the ring ''R''.  Identities (2), (3) represent Leibniz rules for more than two factors, and are valid for any derivation.  Identities (4)–(6) can also be interpreted as Leibniz rules. Identities (7), (8) express '''Z'''-[[Bilinear map|bilinearity]].

From identity (9), one finds that the commutator of integer powers of ring elements is:
: <math>[A^N, B^M] = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1} A^{n}B^{m} [A,B] B^{N-n-1}A^{M-m-1} = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1} B^{n}A^{m} [A,B] A^{N-n-1}B^{M-m-1}</math>

Some of the above identities can be extended to the anticommutator using the above ± subscript notation.<ref>{{harvtxt|Lavrov|2014}}</ref>
For example:
# <math>[AB, C]_\pm = A[B, C]_- + [A, C]_\pm B</math>
# <math>[AB, CD]_\pm = A[B, C]_- D + AC[B, D]_- + [A, C]_- DB + C[A, D]_\pm B</math>
# <math>[[A,B],[C,D]]=[[[B,C]_+,A]_+,D]-[[[B,D]_+,A]_+,C]+[[[A,D]_+,B]_+,C]-[[[A,C]_+,B]_+,D]</math>
# <math>\left[A, [B, C]_\pm\right] + \left[B, [C, A]_\pm\right] + \left[C, [A, B]_\pm\right] = 0</math>
# <math>[A,BC]_\pm = [A,B]_- C + B[A,C]_\pm = [A,B]_\pm C \mp B[A,C]_-</math>
# <math>[A,BC] = [A,B]_\pm C \mp B[A,C]_\pm</math>

==== 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>