Geodesic
The existence of root subgroup translated by a given element into its opposite. II
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 40, Tome 531 (2024), pp. 147-151

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Let $\Phi$ be a simply-laced root system, $|K|>5$, $G = G_{ad}(\Phi,K)$ the adjoint group of type $\Phi$ over $K$. Then for every non-trivial element $g\in G$ there exists a root element $x$ of the Lie algebra of $G$ such that $x$ and $gx$ are opposite. 
@article{ZNSL_2024_531_a8,
     author = {I. M. Pevzner},
     title = {The existence of root subgroup translated by a given element into its opposite. {II}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {147--151},
     publisher = {mathdoc},
     volume = {531},
     year = {2024},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_531_a8/}
}
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I. M. Pevzner. The existence of root subgroup translated by a given element into its opposite. II. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 40, Tome 531 (2024), pp. 147-151. http://geodesic.mathdoc.fr/item/ZNSL_2024_531_a8/