  Jose-Rao introduced and studied the Special Unimodular Vector group
$SUm_r(R)$ and $EUm_r(R)$, its Elementary Unimodular Vector subgroup. They
proved that for $r \geq 2$, $EUm_r(R)$ is a normal subgroup of $SUm_r(R)$. The
Jose-Rao theorem says that the quotient Unimodular Vector group,
$SUm_r(R)/EUm_r(R)$, for $r \geq 2$, is a subgroup of the orthogonal quotient
group $SO_{2(r+1)}(R)/EO_{2(r + 1)}(R)$. The latter group is known to be
nilpotent by the work of Hazrat-Vavilov, following methods of A. Bak; and so is
the former.
  In this article we give a direct proof, following ideas of A. Bak, to show
that the quotient Unimodular Vector group is nilpotent of class $\leq d =
\dim(R)$. We also use the Quillen-Suslin theory, inspired by A. Bak's method,
to prove that if $R = A[X]$, with $A$ a local ring, then the quotient
Unimodular Vector group is abelian.
