Fixed points subgroups by two involutive automorphisms $\sigma, \gamma$ of compact exceptional Lie groups $F_4, E_6$ and $E_7$

TL;DR

This paper determines the structure of subgroups fixed by two involutive automorphisms in compact exceptional Lie groups F_4, E_6, and E_7, extending previous work on related fixed point subgroups.

Contribution

It identifies the group structures of intersections of fixed point subgroups under two involutions in exceptional Lie groups, providing new classifications.

Findings

Group structures of (F_4)^{σ,γ}, (E_6)^{σ,γ}, (E_7)^{σ,γ} determined

Extends previous fixed point subgroup classifications

Provides explicit subgroup types for these intersections

Abstract

For simply connected compact exceptional Lie groups $G = F_4, E_6$ and $E_7$, we consider two involutions $\sigma, \gamma$ and determine the group structure of subgroups $G^{\sigma,\gamma}$ of $G$ which are the intersection $G^\sigma \cap G^{\gamma}$ of the fixed points subgroups of $G^\sigma$ and $G^{\gamma}$. The motivation is as follows. In [1](see the References of this paper), we determine the group structure of $(F_4)^{\sigma, \sigma'}, (E_6)^{\sigma, \sigma'}$ and $(E_7)^{\sigma, \sigma'}$, and in [2](see the References of this paper), we also determine the group structure of $(G_2)^{\gamma, \gamma'}, (F_4)^{\gamma, \gamma'}$ and $(E_6)^{\gamma, \gamma'}$. So, in this paper, we try to determine the type of groups $(F_4)^{\sigma, \gamma}, (E_6)^{\sigma, \gamma}$ and $(E_7)^{\sigma, \gamma}$.