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

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 $(F_4{)}^{\sigma ,{\sigma }^{\prime }},(E_6{)}^{\sigma ,{\sigma }^{\prime }}$ and $(E_7{)}^{\sigma ,{\sigma }^{\prime }}$, and in [2](see the References of this paper), we also determine the group structure of $(G_2{)}^{\gamma ,{\gamma }^{\prime }},(F_4{)}^{\gamma ,{\gamma }^{\prime }}$ and $(E_6{)}^{\gamma ,{\gamma }^{\prime }}$. So, in this paper, we try to determine the type of groups $(F_4{)}^{\sigma ,\gamma },(E_6{)}^{\sigma ,\gamma }$ and $(E_7{)}^{\sigma ,\gamma }$.

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