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We give a classification of irreducible four-dimensional symmetric spaces$G/H$ which admit compact Clifford-Klein forms, where $G$ is the transvectiongroup of $G/H$. For this, we develop a method that applies to particular1-connected solvable symmetric spaces. We also examine a `solvable analogue' ofKobayashi's conjecture for reductive groups and find an evidence that thereductive assumption in Kobayashi's conjecture is crucial.

Four-dimensional compact Clifford-Klein forms of pseudo-Riemannian symmetric spaces with signature $(2, 2)$