  We find relations between real root decompositions of triples of Lie algebras
corresponding to standard compact Clifford-Klein forms, under the assumption
that these triples are not Lie algebra decompositions in the sense of
Onishchik. This enables us to find new classes of homogeneous spaces of simple
real Lie groups which do not admit standard compact Clifford-Klein forms. In
particular, we show that proper R-regular subalgebras of simple real Lie
algebras never generate homogeneous spaces which admit compact standard
Cliffrod-Klein forms.
