Branching Laws for Non-complex Simple Lie Groups of Type $\mathrm{F}_{4}$

Let $G$ be a noncompact non-complex simple Lie group of type $\mathrm{F}_{4}$, i.e., $G = \mathrm{F}_{4(4)}$ or $\mathrm{F}_{4(-20)}$, and $\Gamma \cong (\mathbb{Z}/2\mathbb{Z})^{n}$ a subgroup in the automorphism group $\operatorname{Aut}(G)$ of $G$, which does not contain any Cartan involution. We show that there does not exist an infinite dimensional irreducible representation of $G$ which is algebraically discretely decomposable upon restriction to $G^{\Gamma}$. As a consequence, a conjecture of branching laws for Klein four-symmetric pairs is verified for $\mathrm{F}_{4(4)}$ and $\mathrm{F}_{4(-20)}$. Finally, we compute and list all the Klein four-symmetric pairs for $\mathrm{F}_{4(4)}$ and $\mathrm{F}_{4(-20)}$.