Cartan Pairs and Shared Orbit Pairs

We study a class of pairs of Lie algebras (g, g 1 ) that we call Cartan pairs; here g is semisimple and g 1 is a reductive in g subalgebra.For these pairs, which generalize symmetric ones, we have standardly defined Cartan subspaces, and consequently the set of restricted roots (g, a) .We prove that there are infinitely many interesting nonsymmetric Cartan pairs.Next we prove that every pair of the well known Brylinski-Kostant list of shared orbit pairs is a Cartan pair.As a continuation of the previous research we obtained some further useful and clarifying results and examples related to Cartan pairs and Cartan subspaces.