Elementary subgroup of an isotropic reductive group is perfect

TL;DR

This paper proves that the elementary subgroup of an isotropic reductive algebraic group over a ring is perfect, with exceptions for certain types, clarifying its algebraic structure.

Contribution

It establishes the perfection of the elementary subgroup for most isotropic reductive groups, except for specific known cases, extending previous algebraic group theory results.

Findings

Elementary subgroup is perfect for most isotropic reductive groups.

Exceptions occur for split groups of type C_2 and G_2.

Clarifies the algebraic structure of elementary subgroups.

Abstract

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that the elementary subgroup E(R) of group of points G(R) is correctly defined. Then E(R) is perfect, except for the well-known cases of a split reductive group of type C_2 or G_2.