Elementary subgroups of isotropic reductive groups

Abstract:

Let $G$ be a not necessarily split reductive group scheme over a commutative ring $R$ with $1$. Given a parabolic subgroup $P$ of $G$, the elementary group $\mathrm {E}_P(R)$ is defined to be the subgroup of $G(R)$ generated by $\mathrm {U}_P(R)$ and $\mathrm {U}_{P^-}(R)$, where $\mathrm {U}_P$ and $\mathrm {U}_{P^-}$ are the unipotent radicals of $P$ and its opposite $P^-$, respectively. It is proved that if $G$ contains a Zariski locally split torus of rank 2, then the group $\mathrm {E}_P(R) =\mathrm {E}(R)$ does not depend on $P$, and, in particular, is normal in $G(R)$.