A characterization of the Kostrikin radical of a Lie algebra

In this paper we study if the Kostrikin radical of a Lie algebra is the intersection of all its strongly prime ideals, and prove that this result is true for Lie algebras over fields of characteristic zero, for Lie algebras arising from associative algebras over rings of scalars with no 2-torsion, for Artinian Lie algebras over arbitrary rings of scalars, and for some others. In all these cases, this implies that nondegenerate Lie algebras are subdirect products of strongly prime Lie algebras, providing a structure theory for Lie algebras without any restriction on their dimension.