Relative commutator calculus in Chevalley groups

We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in Chevalley groups G(Φ,R)G(Φ,R), rk(Φ)⩾2rk(Φ)⩾2, which are both more general, and substantially easier than the ones available in the literature. For classical groups such relative commutator calculus has been recently developed by the authors in Hazrat, Zhang (2011) [34], Hazrat et al. (2011) [33]. As an application we prove the mixed commutator formula,[E(Φ,R,a),G(Φ,R,b)]=[E(Φ,R,a),E(Φ,R,b)],[E(Φ,R,a),G(Φ,R,b)]=[E(Φ,R,a),E(Φ,R,b)], for two ideals a,b⊴Ra,b⊴R. This answers a problem posed in a paper by Alexei Stepanov and the second author.