Standard Parabolics Are Classified by Simple Roots
Let \(G\) be a connected reductive group, \(B\subseteq G\) a Borel subgroup, and \(T\subseteq B\) a maximal torus. Let \(\Delta\) be the corresponding set of simple roots. For each subset \(I\subseteq\Delta\), there is a standard parabolic subgroup \(P_I\supseteq B\). The assignment \[ I\longmapsto P_I \] is an inclusion-preserving bijection from subsets of \(\Delta\) to parabolic subgroups of \(G\) containing \(B\).
Every parabolic subgroup of \(G\) is conjugate to exactly one standard parabolic \(P_I\).
Proof
For \(I\subseteq\Delta\), let \(W_I\) be the subgroup of the Weyl group generated by the simple reflections \(s_\alpha\) for \(\alpha\in I\). Define \[ P_I=\bigcup_{w\in W_I} BwB. \] The Bruhat multiplication relations show this union is a closed subgroup containing \(B\); the quotient \(G/P_I\) is complete, so \(P_I\) is parabolic.
Conversely, if \(P\supseteq B\) is parabolic, its image in \(G/B\) is a closed union of Bruhat cells stable under left multiplication by \(B\). The set of Weyl group elements whose cells occur is a parabolic interval generated by a unique subset \(I\subseteq\Delta\). Hence \(P=P_I\). Since every parabolic contains a Borel and all Borels are conjugate, every parabolic is conjugate to a standard one.