Starting from the relation $left[H_{i}, E_{alpha} ight]=alpha_{i} E_{alpha}$ and its adjoint, and using the Hermiticity of the

Starting from the relation $\left[H_{i}, E_{\alpha}\right]=\alpha_{i} E_{\alpha}$ and its adjoint, and using the Hermiticity of the Cartan generators, show that the $E_{\alpha} \mathrm{s}$ are non-Hermitian and appear in pairs with opposite eigenvalues $E_{ \pm \alpha}$, with $E_{\alpha}^{\dagger}=E_{-\alpha}$.