Editing Commutator (section)

==== 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>