In this article we extend the validity of Suslin’s Local-Global Principle for the elementary transvection subgroup of the general linear group GL
n
(
R
)
_n(R)
, the symplectic group Sp
2
n
(
R
)
_{2n}(R)
, and the orthogonal group O
2
n
(
R
)
_{2n}(R)
, where
n
>
2
n > 2
, to a Local-Global Principle for the elementary transvection subgroup of the automorphism group Aut
(
P
)
(P)
of either a projective module
P
P
of global rank
>
0
> 0
and constant local rank
>
2
> 2
, or of a nonsingular symplectic or orthogonal module
P
P
of global hyperbolic rank
>
0
> 0
and constant local hyperbolic rank
>
2
> 2
. In Suslin’s results, the local and global ranks are the same, because he is concerned only with free modules. Our assumption that the global (hyperbolic) rank
>
0
> 0
is used to define the elementary transvection subgroups. We show further that the elementary transvection subgroup ET
(
P
)
(P)
is normal in Aut
(
P
)
(P)
, that ET
(
P
)
=
(P) =
T
(
P
)
(P)
, where the latter denotes the full transvection subgroup of Aut
(
P
)
(P)
, and that the unstable K
1
_1
-group K
1
(
_1(
Aut
(
P
)
)
=
(P)) =
Aut
(
P
)
/
(P)/
ET
(
P
)
=
(P) =
Aut
(
P
)
/
(P)/
T
(
P
)
(P)
is nilpotent by abelian, provided
R
R
has finite stable dimension. The last result extends previous ones of Bak and Hazrat for GL
n
(
R
)
_n(R)
, Sp
2
n
(
R
)
_{2n}(R)
, and O
2
n
(
R
)
_{2n}(R)
.
An important application to the results in the current paper can be found in a preprint of Basu and Rao in which the last two named authors studied the decrease in the injective stabilization of classical modules over a nonsingular affine algebra over perfect C
1
_1
-fields. We refer the reader to that article for more details.