  The porpose of this article is to introduce and investigate properties of a
tool (the a-hyperbolic rank) which enables us to obtain new examples of
homogeneous spaces G/H which admit and do not admit almost compact
Clifford-Klein forms. We achieve this goal by exploring in greater detail the
technique of adjoint orbits developed by Okuda combined with the well-known
conditions of Benoist. We find easy-to-check conditions on G and H expressed
directly in terms of the Satake diagrams of the corresponding Lie algebras, in
cases when G is a real form of a complex Lie group of types A, D or E6. One of
the advantages of this approach is the fact that we don't need to know the
embedding of H into G. Using the a-hyperbolic rank we also show that the
homogeneous space G/H of reductive type of the real form of E6 (of the IV type)
admits compact Clifford-Klein forms if and only if H is compact. Inspired by
the work of Okuda on symmetric spaces G/H we classify all 3-symmetric spaces
admitting almost compact Clifford-Klein forms.
