Finite-dimensionality requires dominance integrality

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with Cartan subalgebra $\mathfrak{h}=\mathbb{C}h$ and simple root $\alpha$, $\alpha (h)=2$, with coroot $h_{\alpha }$ (Coroot of a Lie-algebra root, The special linear Lie algebra sl_2), and the functional $\lambda \in {\mathfrak{h}}^{\ast }$ defined by $\lambda (h)=-1$.

The Axiom of Choice is assumed; it enters through the root-space theory used by [L1] (The Axiom of Choice).

The highest weight of every finite-dimensional irreducible module is dominant integral, that is, $⟨\lambda ,{\alpha }^{\vee }⟩=\lambda (h_{\alpha })\in {\mathbb{Z}}_{\ge 0}$ for every simple root (Finite-dimensional highest weights are dominant integral, Integral, dominant, and strictly dominant weights).

For ${\mathfrak{s}\mathfrak{l}}_2$ the coroot of the root $\alpha$ satisfies $h_{\alpha }=h$, since $\alpha (h)=2$ forces the normalisation $h_{\alpha }=2H_{\alpha }\mathrm{/}\alpha (H_{\alpha })$ with $H_{\alpha }$ dual to $\alpha$; hence for the functional $\lambda$ of the Given line, $⟨\lambda ,{\alpha }^{\vee }⟩=\lambda (h)=-1$. [definition of the coroot]

A finite-dimensional simple highest weight module has a highest weight vector of some weight, and that highest weight is well defined (Highest-weight vectors and modules, Irreducible, completely reducible, and faithful representations).