I am currently working through Carter's book "Simple Groups of Lie Type", trying to explicitly construct the generators of the Chevalley group of the root system $A_1$.
In the context of $A_1$ let $\Phi = \{ a, -a \}$ be the root system and $\Pi = \{ a \}$ the single simple root.
In §4.4 (page 64 in the link above) Carter gives explicit formulas for how the automorphism $x_r(t)$ for $r \in \Phi$ operates on elements of the Chevally basis, in my case $\{h_a\} \cup \{e_a, e_{-a} \}$.
Question 1: I am confused how $x_r(t)$ operates on $e_{-r}$. Carter writes that for all $r \in \Phi$ (!) we have $$x_r(t).e_{-r} = e_{-r} + th_r - t^2 e_{r}$$ This makes sense for $r = a$ but if I consider $r = -a$ and $x_{-a}(t).e_a$ then this leads to a term including $h_{-a}$ (in the formula above) which is not part of the Chevalley basis.
Question 2: Is it really necessary to take $x_r(t)$ for all $r \in \Phi$ to have a generating set of the Chevalley group? Or can a stronger statement be made (i.e. only take $\pm \Pi$ or something?)? I could not find anything in the book so far.
Thanks in advance!