On centrality of \(\operatorname{K}_2\) for Chevalley groups of type \(\mathsf{E}_\ell\) (Q888852)

Let \(R\) be a commutative ring and let \(\Phi\) be a root system of type \(E_\ell\), \(\ell=6\), 7 or 8. The main result is centrality of \(K_2(\Phi,R)\) in the Steinberg group \(\text{St}(\Phi,R)\). This is derived by means of a Quillen-Suslin local-global principle for \(K_2(\Phi,R)\). First it is shown that \(\text{St}(\Phi,R)\) is an amalgamated product of Steinberg groups associated with root subsystems \(\Psi\subset\Phi\) of type \(A_1\) or \(A_3\). If \(I\) is an ideal in \(R\), then the relative Steinberg group \(\text{St}(\Phi,R,I)\) is also shown to be an amalgamated product. This is then used to transfer known results from \(\text{St}(A_4,R,I)\) to \(\text{St}(\Phi,R,I)\). In particular the local-global principle for \(K_2(\Phi,R)\) is established this way. Finally one can invoke the fact that if \(R\) is local, then \(K_2(\Phi,R)\) is generated by symbols.