  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.
