  In our previous joint papers with Roozbeh Hazrat and Alexei Stepanov we
established commutator formulas for relative elementary subgroups in $GL(n,R)$,
$n\ge 3$, and other similar groups, such as Bak's unitary groups, or Chevalley
groups. In particular, there it was shown that multiple commutators of
elementary subgroups can be reduced to double such commutators. However, since
the proofs of these results depended on the standard commutator formulas, it
was assumed that the ground ring $R$ is quasi-finite. Here we propose a
different approach which allows to lift any such assumptions and establish
almost definitive results. In particular, we prove multiple commutator
formulas, and other related facts for $GL(n,R)$ over an {\it arbitrary}
associative ring $R$.
