Suppose $G$ is a semisimple algebraic group of adjoint type. If $(W,S)$ is the Coxeter system and $s\in S$, and $\alpha_s$ is a simple root with corresponding coroot $\alpha_s^\vee$, is it true that $\alpha_s^\vee(-1)=1$?
I know $\alpha_s^\vee(-1)$ is always an element of order dividing $2$ in the torus, but I am wondering if they are easier to deal with in the case where $G$ is of adjoint type.