The yoga of commutators: further applications

In the present paper we describe some recent applications of localisation methods to the study of commutators in the groups of points of algebraic and algebraic-like groups, such as $\mathrm{GL}(n,R)$, Bak's unitary groups $\mathrm{GU}(2l,R,\Lambda)$ and Chevalley groups $G(\Phi,R)$. In particular, we announce multiple relative commutator formula and general multiple relative commutator formula, as well as results on the bounded width of relative commutators in elementary generators. We also state some of the intermediate results as well as some corollaries of these results. At the end of the paper we attach an updated list of unsolved problems in the field.