Let $G$ be a compact lie group and $g$ it's lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) is 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 $Stab(F)\cong T$. But this is not clear at all from the definition of $Stab(F)$ 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.

Thanks in advance