Let $G$ be a complex reductive group, and let $P$ be parabolic, with unipotent radical $U$ and reductive quotient $L=P/U$. A Levi subgroup is a lift of $L$ into $P$.
My question is how do you tell if an element of $P$ lies in some Levi? For example, if $P = B$ is a Borel, then the reductive quotient is a torus, and a Levi is a maximal torus. So in this case, an element lies in a Levi if and only if it is semisimple. So is there a way to generalize this criterion?
The application I have in mind is the following: let $g \in P_1\cap P_2$, for two parabolics (conjugate to each other if necessary). If $g$ lies in a Levi of $P_1$ must it therefore lie in a Levi of $P_2$?