Given a semisimple complex Lie algebra $\frak{g}$ of rank $r$, with Chevally generators $E\_i,F\_i,K\_i$. Let $V$ be a finite dimensional representation of $\mathfrak{g}$ such that each weight space of $V$ is $1$-dimensional. Let $(i\_1,\dots,i\_k)$ be an ordered set of elements of $\{1,\dots,r\}$ (allowing repeats), and let $\{j\_1,\dots,j\_k\}$ be some permutation of $\{1,\dots,r\}$. For $v$ a highest weight of $V$, the elements
$$
F\_{i\_1} F\_{i\_2} \cdots F\_{i\_k}(v),\quad \text { and } \quad F\_{j\_1} F\_{j\_2} \cdots F\_{j\_k}(v),
$$
must have the same weight. Thus by our assumption they must differ by a scalar multiple. Will this scalar multiple always be an element of $\mathbb{Q}$?



Semisimple Lie algebra modules with $1$-dimensional weight spaces
Let me flesh out the answer suggested in the comments. Any complex simple Lie algebra can be obtained as a complexification of a (split) Lie algebra $\mathfrak{g}\_{\mathbb{Q}}$ defined over the field of rational numbers. The irreducible finite dimensional representations of $\mathfrak{g}$ are complexifications of representations of $\mathfrak{g}\_{\mathbb{Q}}$ which in turn can be constructed as quotients of Verma modules. That is
$$
V\_\lambda = \mathfrak{U}(\mathfrak{g}\_\mathbb{Q}) \otimes\_{ \mathfrak{U}(\mathfrak{b}\_\mathbb{Q})} \mathbb{Q}\_\lambda / \text{maximal submodule},
$$
where $\mathbb{Q}\_\lambda$ is the one dimensional representation on which the (split) Cartan subalgebra of $\mathfrak{g}\_\mathbb{Q}$ acts by character $\lambda.$


I am not sure about integrality results which were the subject of the original question.


