Rook placements in \(G_2\) and \(F_4\) and associated coadjoint orbits

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 Andrè 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.