I'm reading about the construction of Chevalley groups over an arbitrary field K.
Carter originally introduces an automorphism on the Chevalley basis $\{h_r,r \in \Pi; \quad e_r, r \in \Phi \}$ and then 'extends' it to act on $\{1_K \otimes h_r , r \in \Pi; 1_K \otimes e_r, r \in \Phi \}$.
He then mentions that when just acting on the normal Chevalley basis, for fixed $r \in \Phi$ we can represent $x_r(\zeta) = \text{exp}(\zeta\text{ad}_{e_r})$ by a matrix with elements of the form $a \zeta^i$ where $a \in \mathbb{Z}$ and $i \geq 0$.
He says the automorphism that acts on the extended Chevalley basis can also be represented by a matrix for fixed root with entries of the form: $\bar{a}t^i$ where $t \in K$ and $\bar{a} \textbf{ is the element of the prime field of } K \textbf{ corresponding to }a \in \mathbb{Z}$.
What exactly is meant by this? Am I right in thinking that this is just $1_K \otimes a$?