An elemental characterization of strong primeness in Lie algebras

In this paper we prove that a Lie algebra L is strongly prime if and only if [x,[y,L]]≠0 for every nonzero elements x,y∈L. As a consequence, we give an elementary proof, without the classification theorem of strongly prime Jordan algebras, of the fact that a linear Jordan algebra or Jordan pair T is strongly prime if and only if {x,T,y}≠0 for every x,y∈T. Moreover, we prove that the Jordan algebras at nonzero Jordan elements of strongly prime Lie algebras are strongly prime.