Thompson-like characterizations of the solvable radical

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x G the subgroup of G generated by x and y is solvable. This confirms a conjecture of Flavell. We present analogues of this result for finite-dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.