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