We find relations between real root decompositions of Lie triples \((\mathfrak {g},\mathfrak {h},\mathfrak {l})\) corresponding to standard compact standard Clifford-Klein forms, under the assumption that \((\mathfrak {g},\mathfrak {h},\mathfrak {l})\) is not a Lie algebra decomposition in the sense of Onishchik. This enables us to find new classes of homogeneous spaces G/H of simple real Lie groups which do not admit standard compact Clifford-Klein forms. In particular, we show that proper R-regular subalgebras of non-compact type \(\mathfrak {h}\) of absolutely simple real Lie algebras \(\mathfrak {g}\) never generate homogeneous spaces G/H which admit compact Clifford-Klein forms.
