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.