Completely reducible Lie algebras of linear transformations

Abstract

W. W. Morozov [10] and [1i]1 has announced and indicated proofs of the following theorems. If 2 is a semi-simple Lie algebra over the field of complex numbers, then any element d of 2 such that Ad d is nilpotent can be imbedded in a three-dimensional simple subalgebra. If 9 is a semi-simple subalgebra of a semi-simple algebra 3 over the field of complex numbers, then the centralizer 9 of V in ? is a direct sum of a semi-simple algebra and its center. Moreover, the elements of the center have adjoint mappings that have simple elementary divisors. The proof of the first result appears to have a gap2 and the proof of the second result is sketched only for the case 9) a three-dimensional simple algebra. In the present note we shall give simple and complete proofs of these results for arbitrary base fields of characteristic 0. Moreover, we shall give these results what appears to be their natural setting, namely, the theory of completely reducible Lie algebras of linear transformations. We shall also extend Morozov's first result to Lie triple systems (Lemma 4) and we shall use this extension to obtain an analogue of this result for Jordan algebras (Theorem 8).