Realizations of globally exceptional $\mathbf{Z}_2 \times \mathbf{Z}_2$-symmetric spaces

Abstract

In [3], a classification is given of the exceptional $\mathbf{Z}_2 \times \mathbf{Z}_2$-symmetric spaces $G/K$, where $G$ is an exceptional compact Lie group or $Spin(8)$, and moreover the structure of $K$ is determined as Lie algebra. In the present article, we give a pair of commuting involutive automorphisms (involutions) $\tilde{\sigma}$, $\tilde{\tau}$ of $G$ concretely and determine the structure of group $G^\sigma \cap G^\tau$ corresponding to Lie algebra $\mathfrak{g}^\sigma\cap \mathfrak{g}^\tau$, where $G$ is an exceptional compact Lie group. Thereby, we realize exceptional $\mathbf{Z}_2 \times \mathbf{Z}_2$-symmetric spaces, globally.