Dominant weights in fundamental coordinates

Given: The Axiom of Choice, such $\mathfrak{g},\mathfrak{h}$ and a chosen base of simple roots ${\alpha }_1,\dots ,{\alpha }_r$ with fundamental weights ${\omega }_1,\dots ,{\omega }_r$.

The Axiom of Choice is assumed; it enters through the root-space theory supplying the real form and the coroot basis of [L1] (The Axiom of Choice).

The simple coroots $h_{{\alpha }_1},\dots ,h_{{\alpha }_r}$ form a basis of $\mathfrak{h}$; the simple roots form a basis of $E={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }$; and the fundamental weights are the dual basis to the simple coroots, $({\omega }_i,{\alpha }_j^{\vee })={\delta }_{ij}$, and form a basis of the weight lattice $P$ (The roots form a reduced crystallographic Euclidean root system, Simple roots form a signed integral basis, Fundamental weights, Root, coroot, weight, and coweight lattices).

$\lambda$ is integral when $⟨\lambda ,{\alpha }_i^{\vee }⟩=\lambda (h_{{\alpha }_i})\in \mathbb{Z}$ for all $i$, dominant integral when all these integers are nonnegative, and every integral functional lies in $E$; for $\lambda \in E$ the expansion in the dual basis is $\lambda ={\sum }_i⟨\lambda ,{\alpha }_i^{\vee }⟩{\omega }_i$ (Integral, dominant, and strictly dominant weights).