Which vectors are weight vectors for some Cartan subalgebra?

Question

Let $\mathfrak g$ be a semisimple Lie algebra and $V$ - its representation, both finite-dimensional over $\mathbb C$. By the well-known results, one can choose in $\mathfrak g$ a Cartan subalgebra and decompose the vector space $V$ as a direct sum of its weight spaces. Elements of the weight spaces are called weight vectors. This decomposition depends on the choice of a Cartan subalgebra. Define a set $\Xi$ to consist of all $x \in V$ such that $x$ is a weight vector for some choice of a Cartan subalgebra. What can be said about the set $\Xi$? I'm asking about topological properties (both with respect to the usual and the Zariski topology), characterizations in terms of explicit equations, if possible finer geometric characteristics (e.g. dimension).