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Have you tried Chapter 17, section 17.1 from Springer's book on algebraic groups? I believe that is as down-to-earth as it can get, and it is certainly rather detailed.

By Lemma 28(b), $H$ is an abelian group generated by the $h_i(t)$'s (where $h_i = h_{\alpha_i}$), and since each $h_i$ is multiplicative (by Lemma 28(a)), the existence follows. To prove uniqueness, …

I am looking for an explicit description of the unipotent radical of a minimal parabolic subgroup of a unitary group, i.e. the group of isometries of a hermitian form, over an arbitrary field. In his …

Yes, this is possible. (Notice that your question implicitly requires $k[G]$ to be an integral domain, otherwise you cannot take its fraction field.) Let $k$ be a non-perfect field of characteristic …

I apologize in advance if my question is too elementary for MO. It is a well known fact that the linear algebraic group $G = \mathsf{SO}_n$ is connected, and there exist a few different proofs of thi …

I assume you mean that there is an isomorphism of varieties $$ (U \dot v \cap \dot v U^-) \times B \to B \dot v B : (x,y) \mapsto xy.$$ Note that it is actually somewhat more natural to write the isom …

It is perhaps easiest to express this in terms of Hopf algebras. The coordinate ring $K[G]$ has the structure of a Hopf algebra; the subvariety $Y$ is a closed subgroup of $G$ if and only if the ideal …

In the case of groups of rank 2, such as your examples $\mathrm{SL}_3(\mathbb{F}_2)$ or $\mathsf{G}_2(3)$, the building is rather easy to describe (either as an incidence geometry or as a bipartite gr …