Let $G$ be a reductive algebraic group and $I$ the set of vertices of the Dynkin diagram of $G$. Let $J \subset I$ and $P_J = BW_JB$ the parabolic subgroup of $G$ containing $B$, where $B$ is a Borel subgroup of $G$ and $W_J$ is the subgroup of the Weyl group $W$ of $G$ generated by $s_j, j \in J$.
Let $L_J$ be the Levi subgroup of $P_J$. Then there is a projection $\pi_J: P_J \to L_J$. Is there a projection from $G$ to $L_J$? Thank you very much.