Question on definition of Weyl Group in Brian Hall

Question

I wanted to follow a proof of Brian Hall "Lie Groups, Lie Algebras and their Representations" but I'm stuck with Definiton 7.25 of the Weyl group which he gives as the group generated by all reflections $s_\alpha:\mathfrak{h}\longrightarrow\mathfrak{h}$ such $$ s_{\alpha}\cdot H=H-\frac{2\left\langle \alpha,\,H\right\rangle }{\left\langle \alpha,\,\alpha\right\rangle }\alpha, $$ Now I'm quite perplexed on this definition... is it right? Should it be $H_\alpha$ instead of $\alpha$? Isn't $\alpha$ in the dual of the cartan subalgebra?