We give a complete classification of irreducible symmetric spaces for which there exist proper SL(2,R)-actions as isometries, using
the criterion for proper actions by T. Kobayashi [Math. Ann. ’89] and combinatorial techniques of nilpotent orbits. In particular, we
classify irreducible symmetric spaces that admit surface groups as discontinuous groups, combining this with Benoist’s theorem
[Ann. Math. ’96].