Results on ${\mathrm{K}}_1$ of general quadratic groups

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.