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 formula_1 is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity with respect to the Lie bracket formula_2, defined for all elements formula_3 in the ring formula_1. The Lie ring formula_1 is defined to be an n-Engel Lie ring if and only if
formula_8 (n copies of formula_9), is satisfied.
In the case of a group formula_10, in the preceding definition, use the definition and replace formula_11 by formula_12, where formula_12 is the identity element of the group formula_10.