I've read that if $\frak g$ is a complex semisimple Lie algebra and $G$ a Lie group with Lie algebra $\frak g$, and $H\in{\frak g}$ is regular semisimple, then the adjoint orbit $${\cal O}_H=\{{\rm Ad}_gH\in{\frak g}:g\in G\}$$ can also be described as the set of all $X\in{\frak g}$ that have the same minimal polynomial as $H$: $${\cal O}_H=\{X\in{\frak g}:m_X=m_H\}.$$ That's an interesting statement. Does anybody know how to prove this? Or do you know a reference where they discuss this?
Certainly, if $X\in{\cal O}_H$ then $X$ has the same minimal polynomial as $H$ since they are similar matrices. But what about the converse?