Distinguished nilpotent orbits, Kostant pairs and normalizers of Lie algebras

A pair of Lie algebras (g,g1) will be called a Kostant pair if g is semisimple, g1 is reductive in g and the restriction of the Killing form Bg to g1 is nondegenerate. We study the class of such (nonsymmetric) pairs and obtain some useful and new structural results. We study the structure of the normalizers Ng(g1), and as a consequence we obtain some corresponding worthy results about algebraic groups. In particular we consider an interesting case when g1 is a distinguished sl2-subalgebra of g. Combined with the research due to V.L. Popov we observe that the notions of self-normalizing (reductive) subalgebras of a semisimple Lie algebra and projective self-dual algebraic subvarieties of the usual nilpotent cones are closely related.