Is $w^{-1} e_{\alpha} w = e_{w^{-1}(\alpha)}$?

Question

Let $\mathfrak{g}$ be a semisimple Lie algebra and $e_{\alpha}$ a root vector, $\alpha$ is a positive root. Let $w \in W$, where $W$ is the Weyl group of $\mathfrak{g}$. Is $w^{-1} e_{\alpha} w = e_{w^{-1}(\alpha)}$? Thank you very much.