  Let $\Phi$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a
commutative ring and let $I,J$ be two ideals of $R$. In the present paper we
describe generators of the commutator groups of relative elementary subgroups
$\big[E(\Phi,R,I),E(\Phi,R,J)\big]$ both as normal subgroups of the elementary
Chevalley group $E(\Phi,R)$, and as groups. Namely, let $x_{\a}(\xi)$,
$\a\in\Phi$, $\xi\in R$, be an elementary generator of $E(\Phi,R)$. As a normal
subgroup of the absolute elementary group $E(\Phi,R)$, the relative elementary
subgroup is generated by $x_{\a}(\xi)$, $\a\in\Phi$, $\xi\in I$. Classical
results due to Michael Stein, Jacques Tits and Leonid Vaserstein assert that as
a group $E(\Phi,R,I)$ is generated by $z_{\a}(\xi,\eta)$, where $\a\in\Phi$,
$\xi\in I$, $\eta\in R$. In the present paper, we prove the following
birelative analogues of these results. As a normal subgroup of $E(\Phi,R)$ the
relative commutator subgroup $\big[E(\Phi,R,I),E(\Phi,R,J)\big]$ is generated
by the following three types of generators: i)
$\big[x_{\alpha}(\xi),z_{\alpha}(\zeta,\eta)\big]$, ii)
$\big[x_{\alpha}(\xi),x_{-\alpha}(\zeta)\big]$, and iii)
$x_{\alpha}(\xi\zeta)$, where $\alpha\in\Phi$, $\xi\in I$, $\zeta\in J$,
$\eta\in R$. As a group, the generators are essentially the same, only that
type iii) should be enlarged to iv) $z_{\alpha}(\xi\zeta,\eta)$. For classical
groups, these results, with much more computational proofs, were established in
previous papers by the authors. There is already an amazing application of
these results, namely in the recent work of Alexei Stepanov on relative
commutator width.
