Compatible pairs of Borel subalgebras and shared orbit pairs

Consider a class of pairs (g,g1), where g is a semisimple Lie algebra and g1 is a subalgebra reductive in g, satisfying the following: For any Cartan subalgebra h1⊆g1 there is a unique Cartan subalgebra h⊆g containing h1. Given such a pair (g,g1) and a Borel subalgebra b1⊆g1 we study a (finite) set SBorg(b1) of all Borel subalgebras b⊆g containing b1. In particular we point at the subclass of pairs (g,g1) when SBorg(b1) is a singleton, for every Borel subalgebra b1⊆g1. As a consequence, for such pairs we relate the corresponding flag varieties B(g) and B(g1). As an interesting class of pairs (g,g1), for which we can apply our results on pairs of Borel subalgebras, we study in detail the pairs considered by R. Brylinski and B. Kostant related to shared orbit pairs.