Involutions on a compact 4-symmetric space of exceptional type (Q902243)

Let \((G/H,\sigma)\) be a \(k\)-symmetric space, that is, \(G/H\) is a homogeneous space and \(\sigma\) is an automorphism of \(G\) such that \(G_o^\sigma\subset H\subset G^\sigma\) and \(\sigma^k=\mathrm{Id}\) while \(\sigma^l\neq \mathrm{Id}\) for any \(l<k\); here \(G^\sigma\) and \(G_o^\sigma\) are the set of fixed points of \(\sigma\) and its identity component, respectively. This work deals with the case where \((G/H,\sigma)\) is 4-symmetric and of exceptional type, and \(H\) is not a centralizer of a toral subgroup of \(G\). The authors classify the involutions \(\tau\) commuting with \(\sigma\), when \(G\) is compact and \(\dim H=1\). This result may lead to the classification of non-compact Riemannian 4-symmetric spaces of exceptional type, as the authors will show in a forthcoming work.