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
.
