Engel-like characterization of radicals in finite dimensional Lie   algebras and finite groups
  A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimensional Lie algebra defined over a field of characteristic zero in similar terms. We suggest a conjectural description of the solvable radical of a finite group as the set of Engel-like elements and reduce this conjecture to the case of a finite simple group. 
Tatiana Bandman, Mikhail Borovoi, Fritz Grunewald, Boris Kunyavskii,   and Eugene Plotkin