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 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
for all x , y in L , the n-Engel identity [ x , [ x , … , [ x , [ x , y ] ] , … ] ] = 0 (n copies of x ), is satisfied.
In the case of a group G , in the preceding definition, use the definition [x,y] = x−1 • y−1 • x • y and replace 0 by 1 , where 1 is the identity element of the group G .
