$\renewcommand{\al}{\alpha} \newcommand{\bra}[1]{\left<#1\right>} \newcommand{\brc}[1]{\left(#1\right)} \newcommand{\Aut}[2]{\operatorname{Aut}_{#1}\brc{#2}} \renewcommand{\C}{\mathbb{C}} \newcommand{\si}{\sigma}$ 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 $H^*$. Again, by nondegenerateness of $(-,-)$, for $\al\in\Phi$ we can define a linear function $s_\al\colon H\to H$ such that $s_\al(h)$ is a unique element of $H$ with the property $\brc{s_\al(h),-}=\si_\al\brc{\brc{h,-}}$.
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 $w_\al(r)^{-1}t_\beta w_\al(r)=s_\al\brc{t_\beta}$ as operators on $V$ for roots $\al\ne\pm\beta$?
Motivation. Want to understand the proof of the first equation in the proof of Lemma 19 in R. Steinberg's Lectures on Chevalley Groups.