Let $(G/H, \sigma)$ be a compact $4$-symmetric space of inner and exceptional type. Suppose that the dimension of the center of $H$ is one and $H$ is not a centralizer of a toral subgroup of $G$. In this paper we shall classify the involution $\tau$ of $G$ satisfying $\tau \circ \sigma = \sigma \circ \tau$.
"Involutions on a compact 4-symmetric space of exceptional type." Osaka J. Math. 52 (4) 1101 - 1125, October 2015.