Realizations of involutive automorphisms $\sigma$ and $G^{\sigma}$ of exceptional linear Lie groups $G$, part I, $G=G_{2}$. $F_{4}$ and $E_{6}$

Abstract

classified involutive automorphisms $\sigma$ of simple Lie algebras $\mathfrak{g}$ and determined the type of the subalgebras $\mathfrak{g}^{\sigma}$ of fixed points. Now for connected exceptional universal linear Lie groups $G$ , we shall find involutive automorphisms $\sigma$ and realize the subgroups $G^{\sigma}$ of fixed points explicitly. In this paper we consider the cases of type $G_{2},$ $F_{4}$ and $E_{6}$ . Our results are as follows. (Results of $E_{7}$ will be soon appeared in this Journal).