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

Let g be a real semisimple Lie algebra with Killing form B and k a B-nondegenerate subalgebra of g of maximal rank. We give a description of all adk-invariant decompositions g=k+m++m− such that B|m±=0, B(k,m++m−)=0 and k+m± are subalgebras. It is reduced to a description of parabolic subalgebras of g with given reductive part k. 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.