Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of simple complex Lie algebra with root system $\Phi$. A subset $D$ the set $\Phi^+$ positive roots is called rook placement if it consists pairwise non-positive scalar products. To each and map $\xi$ from to $\mathbb{C}^{\times}$ nonzero numbers one can naturally assign coadjoint orbit $\Omega_{D,\xi}$ in dual space $\mathfrak{n}^*$. By definition, $f_{D,\xi}$, where $f_{D,\xi}$ sum covectors $e_{\alpha}^*$ multiplied by $\xi(\alpha)$, $\alpha\in D$. (In fact, almost all orbits studied at moment have such form for certain $\xi$.) It follows results Andr\`e that $\xi_1$ $\xi_2$ are distinct maps then $\Omega_{D,\xi_1}$ $\Omega_{D,\xi_2}$ do not coincide classical systems We prove this true $\Phi$ type $G_2$, or $F_4$ orthogonal.