The existence of root subgroup translated by a given element into its opposite

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.