$\renewcommand{\al}{\alpha} \newcommand{\bra}[1]{\left<#1\right>} \newcommand{\brc}[1]{\left(#1\right)} \newcommand{\Aut}[2]{\operatorname{Aut}_{#1}\brc{#2}} \renewcommand{\C}{\mathbb{C}} \newcommand{\si}{\sigma} \newcommand{\brr}[1]{\left[#1\right]}$ Let $L$ be a complex semisimple finite dimensional Lie algebra with a finite dimensional faithful representation $V$. As known, $$ L=H\bigoplus_{\alpha\in\Phi}L_\alpha, $$ where $H$ is a Cartan subalgerba, $\Phi$ is a root system and $L_\alpha=\bra{e_\al}$ are one dimensional root subspaces. Let $\brc{-,-}$ be the Killing form on $L$. Since $(-,-)$ is a nondegenerate symmetric bilinear form on $H$, we can fix an isomorphism $H\to H^*$ by $t_\lambda\mapsto \lambda=\brc{t_\lambda,-}$. Now let $(\lambda,\mu)=\brc{t_\lambda,t_\mu}$ for $\lambda,\mu\in H^*$. This gives a nondegenerate symmetric bilinear form on $H^*$. Let $\si_\al$ be the reflection in the hyperplane orthogonal to $\al$ in $H^*$. Also define $\bra{\lambda,\mu}=2\frac{\brc{\lambda,\mu}}{\brc{\mu,\mu}}$ for $\lambda,\mu\in H^*$.
Since $e_\al$ are nilpotent operators on $V$, for $r\in\C\setminus\{0\}$ we can define operators in $\Aut{}{V}$ by $$ x_\al(r)=\exp\brc{re_\al}=\sum_{i=0}^\infty\frac{r^i}{i!}e_\al^i,\qquad w_\al(r)=x_\al(r) x_{-\al}\brc{-\frac{1}{r}}x_\al(r). $$
The question. Why there is some $c\in\C$ such that $w_\al(r)e_\beta w_\al(r)^{-1}=cr^{-\bra{\beta,\al}}e_{\si_\al\brc{\beta}}$ as operators on $V$ for roots $\al$, $\beta$?
Motivation. Want to understand the proof of the (a) in the proof of Lemma 19 in R. Steinberg's Lectures on Chevalley Groups.
Attempt. In the proof it is said that we should apply
Lemma 19 (b). If $v\in V_\mu$ then there exists $v'\in V_{\si_\al\brc{\mu}}$ independent of $r$ such that $w_\al(r)v=r^{-\bra{\mu,\al}}v'$. Here $V_\lambda$ for $\lambda\in H^*$ are weight subspaces of $V$.
to the adjoint representation of $L$. Then for some $c\in\C$ we have $$w_\al(r)e_\beta w_\al(r)^{-1}=\brc{\brr{w_\al(r),e_\beta}+e_\beta w_\al(r)}w_\al(r)^{-1}=e_\beta+r^{-\bra{\beta,\al}}ce_{\si_\al\brc{\beta}}w_\al(r)^{-1}, $$ by Lemma 19 (b). However, have no idea how to proceed.