Inspired by this question I'd like to ask something more specific:
In the theory of rational points of algebraic connected reductive groups over fields, one often reads about the centraliser of a maximal split $k$-torus. torus. Here is one example: Let 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 $n$, n \ge 1$, in the set $k$-group ${\rm{SL}}_ n(D)$ the $k$-subgroup $T$ consisting of diagonal matrices with entries in $k$ {\rm{GL}}_ 1$ is a maximal split $k$-torus of $SL_n(D)$, k$-torus, while the centraliser $S$ of $T$ consists of the diagonal matrices with entries in $D$. D^{\times}$ (I hope I got this right.) 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?
