Lie algebras with anisotropic Engel subalgebras (Q1117013)

Let L be a finite dimensional Lie algebra over an arbitrary field. There have been a number of investigations concerning the structure of L when conditions are placed on ad x for all \(x\in L\), the Engel Theorem being a prime example. There are the c.n. algebras of Benkart and Osborn, the E.t. algebras of Winter, the ad-semisimple algebras of Farnsteiner and the P.E 1-algebras of the author. The present paper considers a similar condition, when each proper Engel subalgebra contains no non-zero ad-nilpotent elements. The author calls these E.a. Lie algebras. He considers how the class compares to the classes just mentioned. We are led to another class, called X-algebras, in which ad x is either nilpotent or semisimple for each \(x\in L\). He finds the real simple X-algebras and finds that over algebraically closed fields of prime characteristic, every X-algebra is c.n. Further considerations over the latter fields show that \({\mathfrak sl}(2,F)\), \(W_ p(F)\) and \({\mathfrak sl}(3,F)/F\cdot 1\) are the only simple Lie algebras whose inner derivations are either nilpotent or semisimple and these are the only simple E.a. algebras which are either restricted or of a total rank one.