Realizations of inner automorphisms of order 4 and fixed points subgroups by them on the connected compact exceptional lie group $E_8$, Part I

Abstract

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jiménez as the type of Lie algebra. Needless to say, these spaces as homogeneous manifolds are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma}$ of order 4 on $G$ and $H$ is a fixed points subgroup $G^\gamma$ of $G$. In the present article, as Part I, for the connected compact exceptional Lie group $E_8$, we give the explicit form of automorphism $\tilde{\sigma}'_4$ of order 4 on $E_8$ induced by the $C$-linear transformation $\sigma'_4$ of 248-dimensional vector space $𝔢^{C}_{8}$ and determine the structure of the group $(E_8)^{\sigma'_4}$. This amounts to the global realization of one of seven cases with an automorphism of order 4 corresponding to the Lie algebra $𝔥 = 𝔰𝔬(6) \oplus 𝔰𝔬(10)$.