# Levi decomposition for the parabolic subgroups

This question is for the algebraic groups. I find I cannot understand Levi decomposition for the parabolic subgroups well.

Denote the parabolic subgroup is P=LV, L is Levi subgroup. I guess that for the classical group, L is the diagonal element and the left part of it and V is right part of it with all the diagonal elements are 1.

Am I right? If yes, how to show it; if no, please give other interested examples.

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No, $L$ could be a block $L = GL(2) \times GL(1) \subset GL(3)$,i.e, matrix of type $$\begin{pmatrix} * & * & 0 \\ * & * &0 \\ 0 &0 & *\end{pmatrix}$$ and $V$ being matrices of the form

$$\begin{pmatrix} 1 & 0 & * \\ 0 & 1 &* \\ 0 &0 & 1\end{pmatrix}.$$

Being Borel subgroup in GL(n) is being a minimal parabolic subgroup. There, you are right.

For general reductive $G$, this is more difficult.

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