View source for Engel element

An element of a Lie ring or an associative ring for which the inner derivation (cf. [[Derivation in a ring|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|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 [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.K. Amayo,   I. Stewart,   "Infinite-dimensional Lie algebras" , Noordhoff  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Kostrikin,   "Certain aspects of the theory of Lie algebras" , ''Selected problems in algebra and logic'' , Novosibirsk  (1973)  pp. 142–160  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Kostrikin,   "Around Burnside" , Springer  (1989)  (Translated from Russian)</TD></TR></table>