In this article we extend the validity Suslin's Local-Global Principle for
the elementary transvection subgroup of the general linear group, the
symplectic group, and the orthogonal group, where n > 2, to a Local-Global
Principle for the elementary transvection subgroup of the automorphism group
Aut(P) of either a projective module P of global rank > 0 and constant local
rank > 2, or of a nonsingular symplectic or orthogonal module P of global
hyperbolic rank > 0 and constant local hyperbolic rank > 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 is used
to define the elementary transvection subgroups. We show further that the
elementary transvection subgroup ET(P) is normal in Aut(P), that ET(P) = T(P)
where the latter denotes the full transvection subgroup of Aut(P), and that the
unstable K_1-group K_1(Aut(P)) = Aut(P)/ET(P) = Aut(P)/T(P) is nilpotent by
abelian, provided R has finite stable dimension. The last result extends
previous ones of Bak and Hazrat for the above mentioned classical groups.

An important application to the results in the current paper can be found in
the work of last two named authors where they have studied the decrease in the
injective stabilization of classical modules over a non-singular affine algebra
over perfect C_1-fields. We refer the reader to that article for more details.