Levi Decomposition of Parabolic Subgroups
Let \(G\) be a connected reductive group with \(T\subseteq B\), and let \(P_I\) be the standard parabolic attached to \(I\subseteq\Delta\). Then \[ P_I=L_I\ltimes R_u(P_I), \] where \(L_I\) is the connected reductive subgroup generated by \(T\) and the root subgroups \(U_{\pm\alpha}\) for roots in the subsystem generated by \(I\). This \(L_I\) is the unique Levi subgroup of \(P_I\) containing \(T\).
Every parabolic subgroup of \(G\) has a Levi decomposition after conjugating to standard position.
Proof
The standard parabolic \(P_I\) is generated by \(B\) and representatives of the simple reflections in \(I\). Its Lie algebra splits into the toral part, the root spaces in the subsystem generated by \(I\), and the positive root spaces outside that subsystem.
The first two pieces integrate to the reductive subgroup \(L_I\). The remaining positive root spaces integrate to a connected normal unipotent subgroup, the unipotent radical \(R_u(P_I)\). Root-subgroup commutator relations show that multiplication \(L_I\times R_u(P_I)\to P_I\) is an isomorphism of varieties and a semidirect-product decomposition of groups. Uniqueness among Levi subgroups containing \(T\) follows from the root-space description.