Recently, I'm learning group theory, Lie algebra, as a particle physics graduate student.
In the simple algebra topic, I know that by definition, we first need to find out all the commutative Cartan generators $\{H_i\}$. And in the adjoint representation (from now on denoted by mathbb), we need to find all the $|\mathbb{E}_\alpha\rangle$ which are eigenstate by all the $\mathbb{H}_i$, action defined as commutation: $$\mathbb{H}_i|\mathbb{E}_\alpha\rangle:=|[\mathbb{H}_i,\mathbb{E}_\alpha]\rangle=\alpha_i|\mathbb{E}_\alpha\rangle.$$ But I don't know how to prove that for each specific "vector" $\vec{\alpha}=(\alpha_1,\alpha_2...)$, with dimension equal to rank, there is only one such $|\mathbb{E}_\alpha\rangle$, in other words, it's non-degenerate for each $\vec{\alpha}$.
Or, it's equally satisfying if somebody can help prove that for each specific $\vec{\alpha}$, there is a unique linear transformation from generators $\{T_a\}$ to $\mathbb{E}_\alpha$. (Physics students only care about matrix representations.)
Thank you in advance! Please lemme know if I didn't illustrate my question fully.