The normalizer of Chevalley groups of type $\mathrm{E}_6$

We consider the simply connected Chevalley group $G(\mathrm{E}_6,R)$ of type $\mathrm{E}_6$ in a 27-dimensional representation. The main goal is to establish that the following four groups coincide: the normalizer of the Chevally group $G(\mathrm{E}_6,R)$ itself, the normalizer of its elementary subgroup $E(\mathrm{E}_6,R)$, the transporter of $E(\mathrm{E}_6,R)$ in $G(\operatorname{E}_6,R)$, and the extended Chevalley group $\overline G(\mathrm{E}_6,R)$. This is true over an arbitrary commutative ring $R$, all normalizers and transporters being taken in $\mathrm{GL}(27,R)$. Moreover, $\overline G(\mathrm{E}_6,R)$ is characterized as the stabilizer of a system of quadrics. This result is classically known over algebraically closed fields; in the paper it is established that the corresponding scheme over $\mathbb{Z}$ is smooth, which implies that the above characterization is valid over an arbitrary commutative ring. As an application of these results, we explicitly list equations a matrix $g\in\mathrm{GL}(27,R)$ must satisfy in order to belong to $\overline G(\mathrm{E}_6,R)$. These results are instrumental in a subsequent paper of the authors, where overgroups of exceptional groups in minimal representations will be studied.