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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 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

$$[x,[x, \ldots, [x,[x,y]],\ldots]] = 0 (n copies of x), is satisfied.$$

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.

Efim Zelmanov
Engel's theorem

References

Traustason, Gunnar (1993). "Engel Lie-Algebras". Quart. J. Math. Oxford 44 (3): 355–384. doi:10.1093/qmath/44.3.355.
Traustason, Gunnar. "Engel groups (a survey)" (PDF).