  R. Lutz introduced the notion of $\varGamma$-symmetric space as a
generalization of the classical notion of symmetric space in 1981, where
$\varGamma$ is a finite abelian group. In the present article, as
$\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$, we give the automorphisms
$\tilde{\sigma}_3, \tilde{\tau}_3$ of order $3$ on the connected compact
exceptional Lie groups $G=G_2, F_4,E_6$ %and construct
$\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$ as the elements of order $3$
in $\Aut(G)$, explicitly and determine the structure of the group $G^{\sigma_3}
\cap G^{\tau_3}$ using homomorphism theorem elementary. These amount to some
global realizations of exceptional $\varmathbb{Z}_3 \times
\varmathbb{Z}_3$-symmetric spaces $G/K$, where $(G^{\sigma_3} \cap
G^{\tau_3})_0 \subseteq K \subseteq G^{\sigma_3} \cap G^{\tau_3}$.
