Let $G=GL_n$ and $P_J = B W_J B$ a parabolic subgroup of $G$, $J \subset I$, $I=\{1，2，\cdots, n-1\}$.
Let $P_J = U L$ be the Levi decomposition, where $L$ is the Levi subgroup of $I_J$ and $U$ the unipotent radical.
Let $\pi_J: P_J \to L = P_J/U$ be the natural projection. I am trying to under this projection.
Let $$ g = \left(\begin{array}{ccccc} x_{1,1} & x_{1,2} & x_{1,3} & x_{1,4} & x_{1,5}\\ 0 & x_{2,2} & x_{2,3} & x_{2,4} & x_{2,5}\\ 0& x_{3,2} & x_{3,3} & x_{3,4} & x_{3,5}\\ 0& x_{4,2} & x_{4,3} & x_{4,4} & x_{4,5}\\ 0& 0 & 0& 0& x_{5,5} \end{array}\right) \in P_J, $$ and $J=\{2,3,4\}$.
Do we have $\pi_J(g)=a U$, where \begin{align} a=\left(\begin{array}{ccccc} x_{1,1} & 0 & 0 & 0 & 0\\ 0 & x_{2,2} & x_{2,3} & x_{2,4} & 0\\ 0 & x_{3,2} & x_{3,3} & x_{3,4} & 0\\ 0 & x_{4,2} & x_{4,3} & x_{4,4} & 0\\ 0 & 0 & 0 & 0 & x_{5,5} \end{array}\right)? \end{align}
Thank you very much.