Structure of Chevalley groups over rings via universal localization

In the current article we study the structure of a Chevalley group G(R)G(R) over a commutative ring R. We generalize and improve the following results on:•the standard, relative, and multi-relative commutator formulas;•the nilpotent structure of [relative] K1K1;•the bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for the principal congruence subgroup, corresponding to a principal ideal.