In the present paper, which is a direct sequel of our papers [10,11,35] joint with Roozbeh Hazrat, we achieve a further dramatic reduction of the generating sets for commutators of relative elementary subgroups in Chevalley groups. Namely, let $��$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $A,B$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(��,R)$ of type $��$ over $R$. The unrelative elementary subgroup $E(��,A)$ of level $A$ is generated (as a group) by the elementary unipotents $x_��(a)$, $��\in��$, $a\in A$, of level $A$. Its normal closure in the absolute elementary subgroup $E(��,R)$ is denoted by $E(��,R,A)$ and is called the relative elementary subgroup of level $A$. The main results of [11,35] consisted in construction of economic generator sets for the mutual commutator subgroups $[E(��,R,A),E(��,R,B)]$, where $A$ and $B$ are two ideals of $R$. It turned out that one can take Stein---Tits---Vaserstein generators of $E(��,R,AB)$, plus elementary commutators of the form $y_��(a,b)=[x_��(a),x_{-��}(b)]$, where $a\in A$, $b\in B$. Here we improve these results even further, by showing that in fact it suffices to engage only elementary commutators corresponding to {\it one\/} long root, and that modulo $E(��,R,AB)$ the commutators $y_��(a,b)$ behave as symbols. We discuss also some further variations and applications of these results.

18 pages. arXiv admin note: text overlap with arXiv:1811.11263