Fixed points subgroups $G^{\sigma ,{\sigma }^{\prime }}$ by two involutive automorphisms $\sigma$, ${\sigma }^{\prime }$ of exceptional compact Lie group $G$, Part II, $G=E_8$

Abstract

For the simply connected compact exceptional Lie group $E_8$, we determine the structure of subgroup $(E_8{)}^{\sigma ,{\sigma }^{\prime }}$ of $E_8$ which is the intersection $(E_8{)}^{\sigma }\cap (E_8{)}^{{\sigma }^{\prime }}$. Then the space $E_8/(E_8{)}^{\sigma ,{\sigma }^{\prime }}$ is the exceptional ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$- symmetric space of type EVIII-VIII-VIII, and that we give two involutions $\sigma ,{\sigma }^{\prime }$ for the space $E_8/(E_8{)}^{\sigma ,{\sigma }^{\prime }}$ concretely.

16pages