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3 Made formulation more accurate, in terms of k-groups rather than groups of rational points

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?

2 changed title according to a comment

1

Whenever I read "centraliser of maximal torus", I think of...

Inspired by this question I'd like to ask something more specific:

In the theory of rational points of algebraic groups, one often reads about the centraliser of a maximal split $k$-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 $n$, the set $T$ of diagonal matrices with entries in $k$ is a maximal split $k$-torus of $SL_n(D)$, while the centraliser $S$ of $T$ consists of the diagonal matrices with entries in $D$. (I hope I got this right.)

Is this example typical or is it too simple-minded to capture the mysteries of these centralisers?