Klein Four subgroups of Lie Algebra Automorphisms

By calculating the symmetric subgroups $\text{Aut}({\text{fru}}_0{)}^{\theta }$ and their involution classes, we classify the Klein four subgroups $\mathrm{\Gamma }$ of $\text{Aut}({\text{fru}}_0)$ for each compact simple Lie algebra ${\text{fru}}_0$ up to conjugation. This leads to a new approach of classification of semisimple symmetric pairs and ${\text{bbZ}}_2\times {\text{bbZ}}_2$-symmetric spaces. We also determine the fixed point subgroup $\text{Aut}({\text{fru}}_0{)}^{\mathrm{\Gamma }}$.