Centrality of ${\mathrm{K}}_2$-functor revisited

We prove that $\mathrm{S}\mathrm{t}(n,A)$ is a crossed module over $$\mathrm{G}\mathrm{L}$(n, A)$underalocalstablerankconditiononanalgebra$A$ over a commutative ring. Our proof uses only elementary localization techniques in terms of pro-groups and stability results for ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$. We also prove similar result for the Steinberg group associated with any sufficiently isotropic general linear group constructed by a quasi-finite algebra.