Engel identity

The Engel identity, named after Friedrich Engel, is a mathematical equation that is satisfied by all elements of a Lie ring, in the case of an Engel Lie ring, or by all the elements of a group, in the case of an Engel group. The Engel identity is the defining condition of an Engel group.

Formal definition
A Lie ring $$L$$ is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity with respect to the Lie bracket $$[x,y]$$, defined for all elements $$x,y$$ in the ring $$L$$. The Lie ring $$L$$ is defined to be an n-Engel Lie ring if and only if $$ [x,[x, \ldots, [x,[x,y]]\ldots]] = 0$$ (n copies of $$x$$), is satisfied.
 * for all $$x, y$$ in $$L$$, the n-Engel identity

In the case of a group $$G$$, in the preceding definition, use the definition [x,y] = x&minus;1 • y&minus;1 • x • y and replace $$0$$ by $$1$$, where $$1$$ is the identity element of the group $$G$$.