K1 of Chevalley groups are nilpotent

AbstractLet Φ be a reduced irreducible root system and R be a commutative ring. Further, let G(Φ,R) be a Chevalley group of type Φ over R and E(Φ,R) be its elementary subgroup. We prove that if the rank of Φ is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(Φ,R)/E(Φ,R) is nilpotent by abelian. In particular, when G(Φ,R) is simply connected the quotient K1(Φ,R)=G(Φ,R)/E(Φ,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications