Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian subalgebra made of hyperbolic elements); $\mathfrak{a}^+$ an open Weyl chamber of $\mathfrak{a}$; and $a_0$ some element of $\mathfrak{a}^+$.

I would like to have the following property:

**Let $g \in G$. Suppose that $\operatorname{Ad} g$ fixes $a_0$. Then $\operatorname{Ad} g$ fixes the whole subspace $\mathfrak{a}$.**

(In other words, the stabiliser of $a_0$ is equal to the stabiliser of $\mathfrak{a}$, commonly called $L$ or $MA$.)

Since all Cartan subspaces are conjugate to each other, an equivalent statement would be:

**Any two distinct open Weyl chambers in $\mathfrak{g}$ are disjoint.**

This looks like it should be a classical result, but I can neither prove it myself nor find a reference. Can anyone help me?

If the proof is longer than a couple of lines, I would rather quote it than rewrite it, so a reference would be appreciated!