For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by certain elements of the torus normalizer realizes the action of the Weyl group of $\Phi$, the automorphisms corresponding to the symmetries of the Dynkin diagram are not inner. However, it is sometimes possible to realize them as inner automorphisms of certain extended groups, e.g. $GL_n$ for $SL_n$ and $O_{2n}$ for $SO_{2n}$. It can be seen explicitly in matrices for the above cases, but when working with the group of type $\mathsf{E}_6$, it becomes difficult to work with matrices. So I wonder if there is an invariant description?

There is a general construction for extended groups, see [Berman, Moody, Extensions of Chevalley groups. Israel J. Math. 22 (1975), no. 1], but it is not clear, how to describe the desired inner automorphisms in its terms.

For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by certain elements of the torus normalizer realizes the action of the Weyl group of $\Phi$, the automorphisms corresponding to the symmetries of the Dynkin diagram are not inner. However, it is sometimes possible to realize them as inner automorphisms of certain extended groups, e.g. $O_{2n}$ for $SO_{2n}$ (edit: none for $SL$). It can be seen explicitly in matrices for the above cases, but when working with the group of type $\mathsf{E}_6$, it becomes difficult to work with matrices. So I wonder if there is an invariant description?

There is a general construction for extended groups, see [Berman, Moody, Extensions of Chevalley groups. Israel J. Math. 22 (1975), no. 1], but it is not clear, how to describe the desired inner automorphisms in its terms.