  (1) If $R$ is an affine algebra of dimension $d\geq 4$ over
$\overline{\mathbb{F}}_{p}$ with $p>3$, then the group structure on ${\rm
Um}_d(R)/{\rm E}_d(R)$ is nice. (2) If $R$ is a commutative noetherian ring of
dimension $d\geq 2$ such that ${\rm E}_{d+1}(R)$ acts transitively on ${\rm
Um}_{d+1}(R),$ then the group structure on ${\rm Um}_{d+1}(R[X])/{\rm
E}_{d+1}(R[X])$ is nice.
