Bi-isotropic decompositions of semisimple Lie algebras and homogeneous bi-Lagrangian manifolds

Let be a real semisimple Lie algebra with Killing form B and a B-nondegenerate subalgebra of of maximal rank. We give a description of all -invariant decompositions such that , and are subalgebras. It is reduced to a description of parabolic subalgebras of with given reductive part . This is obtained in terms of crossed Satake diagrams. As an application, we get a classification of invariant bi-Lagrangian (or equivalently para-Kähler) structures on homogeneous manifolds G/K of a semisimple group G.