Engel element

An element of a Lie ring or an associative ring for which the inner derivation (cf. Derivation in a ring) determined by it is nilpotent. If all the elements of a finite-dimensional Lie algebra over a field are Engel elements, then the algebra is nilpotent (see Engel theorem). The nilpotency index of the derivation in question is called the Engel index. In general, the set of Engel elements of a Lie algebra is not even a subspace. However, when additional conditions like generalized solvability are imposed on this set, it turns out to be a subalgebra and even an ideal [1]. When there are Engel elements of index 2, the Lie algebra is said to be strongly degenerate. The smallest ideal of a Lie algebra the quotient algebra by which is not strongly degenerate is called the Kostrikin radical.

References

[1]	R.K. Amayo, I. Stewart, "Infinite-dimensional Lie algebras" , Noordhoff (1974)
[2]	A.I. Kostrikin, "Certain aspects of the theory of Lie algebras" , Selected problems in algebra and logic , Novosibirsk (1973) pp. 142–160 (In Russian)
[3]	A.I. Kostrikin, "Around Burnside" , Springer (1989) (Translated from Russian)