Admissible square quadruplets and semisimple symmetric spaces (Q1604330)

Let \(\mathfrak g_{\mathbb C}\) be a complex semisimple Lie algebra and \(\mathfrak g\) be a real form of \(\mathfrak g_{\mathbb C}\). Suppose \(\sigma\) is an involution of \(\mathfrak g\). The \(1\) and \(-1\) eigenspaces of \(\sigma\) give the decomposition \(\mathfrak g=\mathfrak h +\mathfrak q\), \(\mathfrak g_{\mathbb C}=\mathfrak h_{\mathbb C} +\mathfrak q_{\mathbb C}\). The pair \((\mathfrak g, \mathfrak h)\) is then called a locally symmetric pair; if furthermore \(\mathfrak g\) is a compact real form of \(\mathfrak g_{\mathbb C}\) then \((\mathfrak g_{\mathbb C}, \mathfrak h_{\mathbb C})\) is called a complex locally symmetric pair. Now if \(\sigma\) and \(\theta\) are two commuting involutions of a complex semisimple Lie algebra \(\mathfrak g_{\mathbb C}\) with the corresponding decompositions \(\mathfrak g_{\mathbb C}=\mathfrak h_{\mathbb C} +\mathfrak q_{\mathbb C}=\mathfrak k_{\mathbb C} +\mathfrak p_{\mathbb C} \) then we have a quadruplet \((\mathfrak g_{\mathbb C}; \mathfrak h_{\mathbb C}, \mathfrak k_{\mathbb C}; \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\) with complex locally symmetric pairs \((\mathfrak h_{\mathbb C}, \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\) and \((\mathfrak k_{\mathbb C}, \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\). In the present paper the author proves the converse of this results, namely if \((\mathfrak h_{\mathbb C}, \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\) and \((\mathfrak k_{\mathbb C}, \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C}) \) are complex locally symmetric pairs then the involutions \(\sigma\) and \(\theta\) are commuting. Using these results the author describes the quadruplets \((\mathfrak g_{\mathbb C}; \mathfrak h_{\mathbb C}, \mathfrak k_{\mathbb C}; \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\) corresponding to the non-Riemannian locally symmetric pair \((\mathfrak g, \mathfrak h)\).