= Jacobi identity
{c}
{title2=$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$}
{wiki}
The Jacobi identity is the compatibility condition on a <Lie bracket>
$$
[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.
$$
Together with an <Alternating Lie bracket>, it defines a <Lie algebra>. Unlike <associativity>, it relates three cyclic bracketings rather than equating two bracketings. Equivalently, using <antisymmetry of a Lie bracket>,
$$
[x,[y,z]]=[[x,y],z]+[y,[x,z]].
$$
Thus the <linear map> $\operatorname{ad}_x:y\mapsto[x,y]$ is a <derivation of a Lie algebra>: acting on a bracket is the sum of acting on each argument. This makes infinitesimal <symmetries> act consistently on each other's products. It also gives $[\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_{[x,y]}$, the identity that makes the <Adjoint representation of a Lie algebra> a <Lie algebra representation>.
For the <commutator> in an <associative algebra>, the cancellation is directly visible:
$$
\begin{aligned}
[A,[B,C]]&=ABC-ACB-BCA+CBA,\\
[B,[C,A]]&=BCA-BAC-CAB+ACB,\\
[C,[A,B]]&=CAB-CBA-ABC+BAC.
\end{aligned}
$$
Each word occurs once with each sign. This proves the <Jacobi identity> without a restriction on the <characteristic of a field>.
For a computed check in the <sl2 matrix commutator example>,
$$
[H,[E,F]]+[E,[F,H]]+[F,[H,E]]=[H,H]+[E,2F]+[F,2E]=0+2H-2H=0.
$$
Here the last two terms are individually nonzero over <characteristic zero>; the identity records their cancellation. In the <cross-product Lie algebra>, the same cyclic cancellation is the <vector triple product identity>.
The historical connection is unusually concrete: Lie and Engel's volume I expands nested <commutators> of <differential operators> into twelve terms and cancels them cyclically, in Chapter 5, §26, equation (7). See https://arxiv.org/pdf/1003.3202#page=119[Lie and Engel, §26, in Merker's translation, pp. 105–106].
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