Suppose I have a simple, simply connected (linear) algebraic group $\mathcal{G}$ over an algebraically-closed field $k$, which could have any characteristic. In fact, to keep things simple, let's imagine $\mathcal{G}$ is simply-laced. Let $\mathcal{B}$ be a choice of Borel, and $\mathcal{T}$ be a maximal torus for $\mathcal{B}$ (and hence for $\mathcal{G}$). Our $\mathcal{B}$ gives a choice of simple positive roots. I'll write $\mathbb{G}_M$ for the multiplicative group of $k$.

Now, suppose we look at the Dynkin diagram for $\mathcal{G}$, and imagine removing a single node from the diagram; for simplicity let's imagine this doesn't disconnect the diagram. Then I believe we should be able to find a subgroup $\mathbb{G}_M\times\mathcal{G}'$ in $\mathcal{G}$ and a Borel subgroup $\mathcal{B}'$ in $\mathcal{G}'$, such that the inclusion $\mathbb{G}_M\times\mathcal{B}'\hookrightarrow \mathcal{G}$ factors through $\mathcal{B}$; such that the Dynkin diagram for $\mathcal{G}'$ is just the Dynkin diagram for $\mathcal{G}$ with our chosen node removed; and indeed such that when we use the inclusion $\mathcal{G'}\hookrightarrow\mathcal{G}$ to map the simple roots of $\mathcal{G}'$ determined by $\mathcal{B'}$ into roots of $\mathcal{G}$, and hence map the Dynkin diagram of $\mathcal{G}'$ into the Dynkin diagram of $\mathcal{G}$, we get the obvious inclusion of 'the diagram with the node removed' into 'the original diagram'.

Moreover, if instead I did disconnect the diagram, I should get a similar picture, with a subgroup $\mathbb{G}_M\times\mathcal{G}_1\times\dots\times\mathcal{G}_k$ where the groups $\mathcal{G}_k$ have Dynkin diagrams corresponding to the components of the picture I get by removing the node from the original Dynkin diagram.

My question is: a) am I right that this kind of operation is all legal and above board, b) are there any caveats one should be aware of, and c) what are good references to appeal to to make this kind of thing rigorous.

`$_3(k)$`

and look at a Levi subgroup of type GL`$_2(k)$`

which has only analmost directproduct decomposition of the type you want. (Textbooks calledLinear Algebraic Groupsmay be helpful sources for structure of parabolic subgroups relative to Dynkin diagrams.) $\endgroup$ – Jim Humphreys May 20 '10 at 15:35