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

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 $G^\sigma \cap G^{\gamma}$ofthefixedpointssubgroupsof$G^\sigma$and$G^{\gamma}$. The motivation is as follows. In [1](see the References of this paper), we determine the group structure of $(F_4)^{\sigma, \sigma’}, (E_6)^{\sigma, \sigma’}$and$(E_7)^{\sigma, \sigma’}$, and in [2](see the References of this paper), we also determine the group structure of $(G_2)^{\gamma, \gamma’}, (F_4)^{\gamma, \gamma’}$and$(E_6)^{\gamma, \gamma’}$. So, in this paper, we try to determine the type of groups $(F_4)^{\sigma, \gamma}, (E_6)^{\sigma, \gamma}$and$(E_7)^{\sigma, \gamma}$.