Relative and unrelative elementary groups, revisited

Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In the present paper we show that the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$ is generated as a group by the elements of the two following forms: 1) $z_{ij}(ab,c)$ and $z_{ij}(ba,c)$, 2) $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\mathrm{\ne }j\le n$, $a\in A$, $b\in B$, $c\in R$. Moreover, for the second type of generators, it suffices to fix one pair of indices $(i,j)$. This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies that for all associative rings one has the equality $[E(n,R,A),E(n,R,B)]=[E(n,A),E(n,B)]$ and many further corollaries can be derived for rings subject to commutativity conditions.