Let $\mathfrak g$ be a compact simple real Lie algebra and let $x\in\mathfrak g$.

What is the intersection of all maximal abelian subalgebras of $\mathfrak g$ which contain $x$?

Let $\mathfrak g$ be a compact simple real Lie algebra and let $x\in\mathfrak g$.

What is the intersection of all maximal abelian subalgebras of $\mathfrak g$ which contain $x$?

For instance, in the simplest case of $\mathfrak{so}(3)$, with $x\neq 0$, the only maximal torus containing $x$ is the 1-dimensional space spanned by $x$. In Lie algebras of higher rank, there are in general multiple maximal tori containing a given $x$.