$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a compact Lie group and $\g$ its Lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) can be identified to a coadjoint orbit. However it is not at all clear to me how one can show this result. I guess we must somehow prove that there always exists an $F\in \g^*$ such that $\operatorname{Stab}(F)\cong T$. But this is not clear at all from the definition of $\operatorname{Stab}(F)$ so I guess the proof for that fact must be using more elegant ideas than that. So can anyone explain what ideas are used to show this or reference somewhere?