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By definition a torus is a group that is isomorphic to $\mathbf{G}_m^n$ for some $n$.

Now given a parabolic subgroup $P$ of a reductive group $G$ (you may assume that $G$ is split and that $P$ is standard with respect to a choice of a Borel if you want) is there a natural decomposition of $L_P$, the Levi subgroup of $P$, as a product of reductive groups which in the case of $P=B$ gives back $T = \mathbf{G}_m^n$ ?

I know this is the case for $G = GL_n$ in which case there is a canonical decomposition $L_p = \prod_{i=1}^r GL_{n_i}$

If yes what kind of information can we get on this decomposition. For examples in the case of $GL_n$ is there a way to find the $n_i$'s in term of data associated to $P$ ?

I apologize for the vagueness of the question but I don't really know what I want to know !

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    What do you mean with "the data associated to $P$"? $P$ stabilizes a flag: if you know the dimensions of the spaces of the flag, then you know the $n_i$'s. More in general, in terms of the associated Lie algebras, if you know which simple roots of $\mathfrak{gl}_n$ are contained in the Levi subalgebra, then you can reconstruct the $n_i$'s. If you need it, I can give some details. – 2017-02-08
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    Thanks for your comment ! I would be very interested in the description in terms of the roots. But the case of GL_n isn't really what I'm after. I am really interested in the case of a general (split) reductive group. – 2017-02-08

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