Abstract:
Let $G=\widetilde G(K)$ where $\widetilde G$ is a simple and simply connected algebraic group that is defined and quasi-split over a field $K$. We consider commutators in $G$ with some regular elements. In particular, we prove (under some additional condition) that every unipotent regular element of $G$ is conjugate to a commutator $[g,v]$, where $g$ is any fixed semisimple regular element of $G$, and that every non-central element of $G$ is conjugate to a product $[g,\sigma][u_\mathrm{reg},\tau]$, where $g$ is some special element of the group $G$ and $u_\mathrm{reg}$ is some regular unipotent element of $G$.
Key words and phrases:commutators in Chevalley groups, regular elements in Chevalley groups, the Ore's problem.