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.
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
is defined to be an n-Engel Lie ring if and only if
is the identity element of the group
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