  Let $R$ be an exchange ring. We prove that the relative elementary subgroups
$E_n(R,I)$ are normal in the general linear group $GL_n(R)$ if $n\geq 1$ and
that the standard commutator formula
$E_n(R,I)=[E_n(R),E_n(R,I)]=[E_n(R),C_n(R,I)]$ holds if $n\geq 3$. Moreover, we
classify the subgroups of $GL_n(R)$ that are normalised by the elementary
subgroup $E_n(R)$ in the case $n\geq 3$.
