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⊂𝔤,𝔩⊂𝔤, 𝔤 is simple and absolutely simple, 𝔥,𝔩 are (non-compact) reductive, 𝔤=𝔥+𝔩, and the intersection 𝔥∩𝔩 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