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.