Let $\mathfrak g$ be a complex simple Lie algebra. Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system. Pick a partial order on $\mathfrak h$, which induces a positive root system $\Delta^+$, fundamental Weyl chamber, etc.

I wish a reference for the following result, which must be well known. My proof is case-by-case and a bit long.

For all but finitely many finite dimensional irreducible representations $(\pi, V)$ of $\mathfrak g$, there is a weight $\mu$ of $\pi$ lying in the fundamental Weyl chamber.

Recall that $\mu\in\mathfrak h^*$ is called a weight of $\pi$ if $V(\mu):=\{v\in V: \pi(X)\cdot v=\mu(X)\, v\quad\forall \, X\in\mathfrak h\}$ is non-zero. Furthermore, the fundamental Weyl chamber is given by elements $\mu\in\mathfrak h^*$ satisfying $\langle \mu,\alpha\rangle >0$ for all $\alpha\in\Delta^+$.

The proof is immediately reduced to consider irreducible representations of $\mathfrak g$ with highest weight lying in a face of the fundamental Weyl chamber. Furthermore, it reduces to check irreducible representations with highest weight a multiple of a fundamental weight.

It is easy to see that the result is no longer valid when $\mathfrak g$ is assumed semisimple in place of simple.