In this paper we deduce a graded version of Quillen--Suslin's Local-Global
Principle for the traditional classical groups, viz. general linear, symplectic
and orthogonal groups and establish its equivalence of the normality property
of the respective elementary subgroups. This generalizes previous result of
Basu--Rao--Khanna. Then, as an application, we establish an analogue
Local-Global Principle for the commutator subgroups of the special linear and
symplectic groups. Finally, by using Swan-Weibel's homotopy trick, we establish
graded analogue of the Local-Global Principle for the transvection subgroups of
the full automorphism groups of the linear, symplectic and orthogonal modules.
This generalizes the previous result of Bak--Basu--Rao.
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