  In the first part of this article we discuss the relative cases of
Quillen-Suslin's local-global principle for the general quadratic (Bak's
unitary) groups, and its applications for the (relative) stable and unstable
$\mathrm{K}_1$-groups. The second part is dedicated to the graded version of
the local-global principle for the general quadratic groups and its application
to deduce a result for Bass' nil groups.
