Realizations of globally exceptional \(\mathbb {Z}_2 \times \mathbb{Z}_2\)-symmetric spaces (Q2340438)

A homogeneous space \(G/K\) is called \((\mathbb{Z}_2\times \mathbb{Z}_2)\)-symmetric if \(G\) acts almost effectively on \(G/K\) and there is an injective homomorphism \(\rho: \mathbb{Z}_2\times \mathbb{Z}_2 \longrightarrow\mathrm{Aut}(G)\), such that \[ G_0^{\mathbb{Z}_2\times \mathbb{Z}_2}\subset K \subset G_0^{\mathbb{Z}_2\times \mathbb{Z}_2}\,. \] Here \({G^{\mathbb{Z}_2\times \mathbb{Z}_2}}\) denotes the subgroup of elements fixed by \(\mathbb{Z}_2\times \mathbb{Z}_2\) and \(G_0^{\mathbb{Z}_2\times \mathbb{Z}_2}\) its identity component. Following earlier work of A.~Kollross, who gave a complete description and classification of \(\mathbb{Z}_2\times \mathbb{Z}_2\) symmetric spaces, the author provides an explicit and detailed description of \(\mathbb{Z}_2\times \mathbb{Z}_2\)-symmetric spaces, associated to the five types of exceptional Lie groups by providing a detailed construction of the quotient subgroups \(K\).