We prove constructively that for any finite-dimensional commutative ring
R
R
and
n
≥
dim
(
R
)
+
2
n\geq \dim (R)+2
, the group
E
n
(
R
[
X
,
X
−
1
]
)
\mathrm {E}_{n}(R[X,X^{-1}])
acts transitively on
U
m
n
(
R
[
X
,
X
−
1
]
)
\mathrm {Um}_{n}(R[X,X^{-1}])
. In particular, we obtain that for any finite-dimensional ring
R
R
, every finitely generated stably free module over
R
[
X
,
X
−
1
]
R[X,X^{-1}]
of rank
>
dim
R
>\dim R
is free; i.e.,
R
[
X
,
X
−
1
]
R[X,X^{-1}]
is
(
dim
R
)
(\dim R)
-Hermite.