Editing Jacobi identity (section)

== Related identities ==

* The [[Commutator#Identities (group theory)|Hall–Witt identity]] is the analogous identity for the [[commutator]] operation in a [[group (mathematics)|group]].

* The following higher order Jacobi identity holds in arbitrary Lie algebra:<ref>{{cite arXiv
 | first1=Ilya |last1=Alekseev | first2=Sergei O. |last2=Ivanov |eprint=1604.05281
 | title       = Higher Jacobi Identities |date=18 April 2016
 |class=math.GR }}</ref>

: <math>[[[x_1,x_2],x_3],x_4]+[[[x_2,x_1],x_4],x_3]+[[[x_3,x_4],x_1],x_2]+[[[x_4,x_3],x_2],x_1] = 0.</math>

* The Jacobi identity is equivalent to the [[Product Rule]], with the Lie bracket acting as both a product and a derivative:  <math>[X,[Y,Z]] = [[X,Y], Z] + [Y, [X,Z]]</math>.   If <math>X, Y</math> are vector fields, then <math>[X,Y]</math> is literally a derivative operator acting on <math>Y</math>, namely the [[Lie derivative]] <math>\mathcal{L}_X Y</math>.