# Engel identity

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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 is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity with respect to the Lie bracket , defined for all elements in the ring . The Lie ring is defined to be an n-Engel Lie ring if and only if

(n copies of ), is satisfied.^{[1]}

In the case of a group , in the preceding definition, use the definition [*x*,*y*] = *x*^{−1} • *y*^{−1} • *x* • *y* and replace by , where is the identity element of the group .^{[2]}