An elemental characterization of strong primeness in Lie algebras

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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.