Local-global principle for general quadratic and general Hermitian groups and the nilpotence of $\mathrm{KH}_1$

In this article we establish an analog of the Quillen–Suslin's local-global principle for the elementary subgroup of the general quadratic group and the general Hermitian group. We show that unstable $\mathrm K_1$-groups of general Hermitian groups over module finite rings are nilpotent-by-abelian. This generalizes earlier results of A. Bak, R. Hazrat, and N. Vavilov.