In the proof of Proposition 7.31 in Roger Carter's Lie Algebras of Finite and Affine Type, Carter notes that the sets $H_\mu$ and $H_\alpha$ are distinct. Can someone find a good argument why that is the case?
I have included the page in question below:
Fix $\alpha \in \Phi$. As $\dim_{\mathbb{R}} H_{\mathbb{R}} = \ell < \infty$ and $\alpha \neq 0$, one can choose elements $\lambda_1, \ldots, \lambda_{\ell-1} \in H^*_{\mathbb{R}}$ such that $\alpha, \lambda_1, \ldots, \lambda_{\ell-1}$ form a basis of $H^*_{\mathbb{R}}$ over $\mathbb{R}$. Write $\mu$ in the form $\mu = c_0\alpha + c_1\lambda_1 + \cdots + c_{\ell-1}\lambda_{\ell-1}$ with $c_0,\ldots,c_{\ell-1} \in \mathbb{R}$. Since $\mu$ is not a multiple of $\alpha$, we have $c_j \neq 0$ for some $j$ with $1 \leq j \le \ell-1$. Denoting the basis of $H_{\mathbb{R}}$ dual to $(\alpha,\lambda_1,\ldots,\lambda_{\ell-1})$ by $(\lambda_0^*, \lambda_1^*,\ldots,\lambda_{\ell-1}^*)$, observe that $c_j\lambda_0^* - c_0\lambda_j^* \in H_{\mathbb{R}}$ belongs to $H_\mu$ but not to $H_\alpha$. Hence, $H_\mu \neq H_\alpha$.