Recently, I began to read a lecture notes, download from internet. The lecture is about finite simple groups. The first interesting thing in this note is about the maximal abelian subgroups $A$ of $G=SL\_2(F)$, where $F$ is a field of characteristic $p\ge 0$: 1) $A$ is a unipotent radical of the Borel subgroup; 2)
$A$ is a maximal split torus; 3) $A$ is a maximal non-split torus.


The first two types are easy to understand by matrix, but the third type is complex to me, and the lecture is not clear about this. Is there anyone give me some detail for this, or give me some references for this?


I also find that there are maybe some trick in the computation of matrix, which can not be found in the course of linear algebra. For example, is there special way to computing $[A,B]$ or oder of $A$ for $A, B \in GL\_n(F)$?


Thank you very much!



About the maximal abelian subgroups of $SL_2(F)$
Here, is another point of view.


Take any elliptic element $\gamma$, i.e. with irreducible characteristic polynomial, then the centralizer is a maximal non split torus.


In fact, the centralizer is $F[\gamma]^\times$.


