Nilpotency of locally isotropic ${\mathrm{K}}_1$-functor

Abstract

We show that the ${\mathrm{K}}_1$-functor modeled on locally isotropic reductive groups is hypoabelian if the base ring has finite Bass--Serre dimension and, for the Tits index ${\mathsf{E}}_{8,2}^{78}$, contains a field. For classical or globally isotropic reductive groups schemes ${\mathrm{K}}_1$ is actually solvable. This implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.