A solution of the problem of standard compact Clifford-Klein forms

We solve the long standing problem of classification of standard compact Clifford-Klein forms of homogeneous spaces of simple non-compact real Lie groups. The result is that standard compact Clifford-Klein forms always arise from triples (g,h,l) of real Lie algebras such that h,l, g is simple and absolutely simple, h,l are (non-compact) reductive, g=h+l, and the intersection h is compact. The consequence of this is the following characterization of proper co-compact actions of reductive Lie subgroups L⊂ G on a homogeneous spaces $G/H$ determined by absolutely simple real Lie group G and a closed reductive subgroup H: L acts on $G/H$ properly and co-compactly if and only if G=H· L and H∩ L is compact