A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic variety. A subgroup $P\subseteq G$ is a parabolic subgroup if and only if it contains some Borel subgroup of the group $G$. Each parabolic subgroup $P$ of a group $G$ is the semi-direct product of its unipotent radical $U$ and the Levi subgroup of the group $P$.

I am interesting in knowing about the dimension of $U$. In particular, for what $G$ can one find a parabolic $P$ whose unipotent subgroup $U$ has dimension a multiple of 4? The answer for $G=\mathrm{SL}_n(\mathbb{R})$ is clear. Is there a discussion of the computations for $U$ in a reference book? Is this something that LiE or Sage can help me determine?

(Apologies if this question has been asked before.)

realsplit semisimple Lie or algebraic group as an example, but it's usually best to focus first on the corresponding complex group or better yet on its Lie algebra. All the dimension information you want is found in the latter more elementary setting. In any case, no computer calculation is needed, just a lot of routine elementary arithmetic. Is there some motivation? $\endgroup$ – Jim Humphreys Jan 19 '13 at 22:04`$\mathbb{Q}$`

includes the ones originally discussed. For them it's easy to compute dimensions of unipotent radicals. For other isotropic but nonsplit forms, you have to dig more into the classification. $\endgroup$ – Jim Humphreys Feb 12 '13 at 1:57