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R. Lutz introduced the notion of $\varGamma$-symmetric space as ageneralization 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 compactexceptional 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 someglobal realizations of exceptional $\varmathbb{Z}_3 \times\varmathbb{Z}_3$-symmetric spaces $G/K$, where $(G^{\sigma_3} \capG^{\tau_3})_0 \subseteq K \subseteq G^{\sigma_3} \cap G^{\tau_3}$.

On some realizations of globally exceptional $\varmathbb{Z}_3 \times \varmathbb{Z}_3 $-symmetric spaces $G/K$, $G=G_2, F_4, E_6$, Part I