TITLE: Whenever I read "centraliser of maximal split torus", I think of...
QUESTION [7 upvotes]: Inspired by this question
I'd like to ask something more specific: 
In the theory of connected reductive groups over fields, one often reads about the centraliser of a maximal split torus. Here is one example: let $k$ be a field and $D$ a skewfield containing $k$ such that $D$ is a finite-dimensional central simple $k$-algebra. Then for any $n \ge 1$, in the $k$-group ${\rm{SL}}_ n(D)$ the $k$-subgroup $T$ consisting of diagonal matrices with entries in ${\rm{GL}}_ 1$ is a maximal split $k$-torus, while the centraliser $S$ of $T$ consists of the diagonal matrices with entries in $D^{\times}$ (viewed as a $k$-group in the usual way). 
Is this example typical or is it too simple-minded to capture the mysteries of these centralisers?

REPLY [5 votes]: The centralizer of a max. split torus is (as Loren noted) the anisotropic
kernel of G.  Maybe the following additional example is useful: Let k be a 
field and Q a non-degenerate quadratic form over k, and let G = SO(Q) (let's 
avoid char. 2 for simplicity...) 
Then [Witt's Theorem] Q can be decomposed into an (orthogonal) sum
Q = Q_an + Q_hyp
where Q_an is an anisotropic quadratic form (has no non-trivial zeros), and
where the quadratic from Q_hyp is hyperbolic ("looks like a quadratic form over
an alg. closed field").
The choice of a hyperbolic basis for Q_hyp is (almost) the same as a choice
of maximal split torus.
And the derived group of the centralizer of that maximal split torus is
the anisotropic group SO(Q_an). [For detail on all this see e.g. [Borel, 
Linear Algebraic Groups 23.4] I'm sure there is an analogous reference
in [Springer, LAG] but my copy of that book is elsewhere at the moment].
Of course, this is similar in spirit to your division algebra example.
For a more elaborate source of examples, see the references Jim cites.