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

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