Realization of globally exceptional Riemannian $4$-symmetric space $E_8/(E_8)^{\sigma'_{4}}$

Abstract

The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{\'{e}}nez. As homogeneous manifolds, these spaces are of the $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma}$ of oder 4 and $H$ is a fixed points subgroup $G^\gamma$ of $G$. In the present article, for the exceptional compact Lie group $G=E_8$, we give the explicit form of automorphism of order 4 induced by the $\boldsymbol{R}$-linear transformation $\sigma'_{4}$ and determine the structure of the group $(E_8)^{\sigma'_{4}}$. Thereby, we realize the globally exceptional Riemannian $4$-symmetric space $E_8/(E_8)^{\sigma'_{4}}$.