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 ad_k-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.