Geodesic
The existence of root subgroup translated by a given element into its opposite
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 32, Tome 460 (2017), pp. 190-202 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\Phi$ be a simply-laced root system, $K$ an algebraically closed field, $G=G_\mathrm{ad}(\Phi,K)$ 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. 
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I. M. Pevzner. The existence of root subgroup translated by a given element into its opposite. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 32, Tome 460 (2017), pp. 190-202. http://geodesic.mathdoc.fr/item/ZNSL_2017_460_a8/

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