Geodesic
Orbits of vectors in some representations. II
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 39, Tome 522 (2023), pp. 125-151

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Let $\Phi$ be a root system of type $A_l$ or $D_l$. Let $K$ be a field of characteristic other than $2$. Let $\delta$ be the maximal root of $\Phi$ and $\Phi_0 = \{\alpha\in\Phi; \delta\perp\alpha\}$. The orbits of the group $G_{\mathrm sc}(\Phi_0, K)$ acting on the set $\langle e_\alpha; \angle(\alpha, \delta) = \pi/3\rangle$ are described. 
@article{ZNSL_2023_522_a7,
     author = {I. M. Pevzner},
     title = {Orbits of vectors in some representations. {II}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {125--151},
     publisher = {mathdoc},
     volume = {522},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_522_a7/}
}
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I. M. Pevzner. Orbits of vectors in some representations. II. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 39, Tome 522 (2023), pp. 125-151. http://geodesic.mathdoc.fr/item/ZNSL_2023_522_a7/