How does Weyl group acts on coroots?

Question

We know that $W=N/T$ ($N = \{n \in G \mid nTn^{-1} = T \}$) acts on $T$ by $w(t)=wtw^{-1}$ (since $T$ is commutative, this action is well-defined). In the case of $GL_n$, by direct computation we know that $w_0(\alpha_i^{\vee}(c))=\alpha_{i^*}^{\vee}(c^{-1})$, where $w_0$ is the longest element in $W$, $\alpha_i^{\vee}$ is a coroot, $i^*$ is $\tau$ and $\tau$ is the automorphism of the Dynkin diagram. My question is how to prove $w_0(\alpha_i^{\vee}(c))=\alpha_{i^*}^{\vee}(c^{-1})$ in other types other than type A (in type A, $GL_n$ case we can use matrices to verify). It is said that this is well-know. But I am not able to find a reference. Are there some references about this fact? Thank you very much.