Orbits of real semisimple Lie groups on real loci of complex symmetric spaces

Let \(G\) be a complex semisimple algebraic group and \(X\) be a complex symmetric homogeneous \(G\)-variety. Assume that both \(G\), \(X\) as well as the \(G\)-action on \(X\) are defined over real numbers. Then \(G(\mathbb{R})\) acts on \(X(\mathbb{R})\) with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of A.Borel and L.Ji.