Let $G$ be a reductive group over $\mathbb{Q}$ and let $P\_0$ minimal parabolic subgroup.



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> If $P\_1=M\_{1}N\_{1} \subset P\_2=M\_2N\_2$ are standard parabolic subgroups of $G$, then can we decompose $P\_1=(P\_1 \cap M\_2 )N\_2$? 
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It seems it does hold but I don't know why it holds.
Any comments are welcome!



Decomposition of parabolic subgroup in reductive group
Over an algebraically closed field (of any characteristic), it is fairly obvious using the omitted definition of "standard parabolic" that the assertion here for a pair of included standard parabolics is true: one just wants to trace the pairs of positive and negative roots involved. However, the question is too loosely formulated to be clear. For example, the minimal $k$-parabolic $P\_0$ is introduced but then ignored for $k=\mathbb{Q}$. Here $k$ may be arbitrary, but it needs to be clarified what "standard" means in this setting. The 1965 foundational paper by Borel and Tits is now somewhat old-fashioned in language, but the BN-pair setting for parabolics is probably helpful here. 


