Product structures and decompositions of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-symmetric spaces

In this paper, we establish the decomposition and product theory of $\mathrm{\Gamma }$-symmetric spaces, where $\mathrm{\Gamma }$ is isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. These results lead to the local classification of semisimple Riemannian ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-symmetric spaces of nontrivial type. Furthermore, we identify notable phenomena specific to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-symmetric spaces: the existence of distinct product structures and that of an example with the trivial infinitesimal isotropy representation.