K ₁ of Chevalley groups are nilpotent

Let 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 K ₁ (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 C ₁ and D ₁ . As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.