Geodesic
Orbits of vectors in some representations. III
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 39, Tome 522 (2023), pp. 152-163

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Let $\Phi$ be a root system of type $E_6$, $E_7$, or $E_8$. Let $K$ be a field of characteristic $2$. Let $\delta$ be the maximal root of $\Phi$ and set $\Phi_0 = \{\alpha\in\Phi; \delta\perp\alpha\}$. 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_a8,
     author = {I. M. Pevzner},
     title = {Orbits of vectors in some representations. {III}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {152--163},
     publisher = {mathdoc},
     volume = {522},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_522_a8/}
}
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I. M. Pevzner. Orbits of vectors in some representations. III. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 39, Tome 522 (2023), pp. 152-163. http://geodesic.mathdoc.fr/item/ZNSL_2023_522_a8/