Editing Jacobi identity (section)

== Definition ==
Let <math>+</math> and <math>\times</math> be two [[binary operation]]s, and let <math>0</math> be the [[neutral element]] for <math>+</math>. The '''{{visible anchor|Jacobi identity}}''' is

:<math>x \times (y \times z) \ +\ y \times (z \times x)  \ +\ z \times (x \times y)\ =\ 0.</math>

Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form <math>a \times (b \times c)</math>, the variables <math>x</math>, <math>y</math> and <math>z</math> are permuted according to the cycle <math>x \mapsto y \mapsto z \mapsto x</math>. Alternatively, we may observe that the ordered triples <math>(x,y,z)</math>, <math>(y,z,x)</math> and <math>(z,x,y)</math>, are the [[Permutation#Parity of a permutation|even permutations]] of the ordered triple <math>(x,y,z)</math>.