Inclusion among commutators of elementary subgroups

In the present paper we continue the study of the elementary commutator subgroups $[E(n,A),E(n,B)]$, where $A$ and $B$ are two-sided ideals of an associative ring $R$, $n\ge 3$. First, we refine and expand a number of the auxiliary results, both classical ones, due to Bass, Stein, Mason, Stothers, Tits, Vaserstein, van der Kallen, Stepanov, as also some of the intermediate results in our joint works with Hazrat, and our own recent papers [40,41]. The gimmick of the present paper is an explicit triple congruence for elementary commutators $[t_{ij}(ab),t_{ji}(c)]$, where $a,b,c$ belong to three ideals $A,B,C$ of $R$. In particular, it provides a sharper counterpart of the three subgroups lemma at the level of ideals. We derive some further striking corollaries thereof, such as a complete description of generic lattice of commutator subgroups $[E(n,I^r),E(n,I^s)]$, new inclusions among multiple elementary commutator subgroups, etc.