About the maximal abelian subgroups of $SL_2(F)$ 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!
 A: The construction of a maximal non-split torus has always felt trickier to me than the other two you mention, but it isn't too hard to understand. I learned most of this by reading Cédric Bonnafé's text Representations of $SL_2(\mathbb{F}_q).$
If we take $K / F$ to be a field extension of degree two, then we can regard $K$ as a two dimensional vector space over $F$, and as multiplication by elements of $K$ is an $F$-linear operation, by choosing a basis of $K$ as a vector space over $F$ we get that $K^\times$ is isomorphic to some subgroup of $GL_2(F)$. Through this isomorphism, the trace and norm of the field extension correspond respectively to the trace and determinant of the matrices. This shows that the elements of $K$ which are of norm 1 over $F$ can be identified with a subgroup of $SL_2(F)$, this subgroup is a maximal non-split torus.
For me the problem with this construction is that it doesn't necessarily make it easy to see what the matrices in this non-split torus look like. What helps me is to try and understand them through analogy with $SL_2(\mathbb{R})$, because there I know a field extension of degree 2 that I understand well. Using 1 and $i$ as our basis for $\mathbb{C}$ and tracing through the isomorphisms there gives us that our non-split torus is $SO_2(\mathbb{R})$ as a subgroup of $SL_2(\mathbb{R})$ in the usual way. While I feel I can get a good mental grasp on this group, it still feels quite a bit more complicated to me than the standard split torus.
I hope you find this useful, It is treated fairly well in Bonnafé's aforementioned text, all in chapter 1, section 1.1.2. There are some errors in some later calculations in the text, but the section on the non-split torus is good, and there is a good exercise on calculating a good basis for $\mathbb{F}_{q^2}$ over $\mathbb{F}_q$ for calculating the matrices you get in the non-split torus.
A: If the characteristic of the field is not $2$, the description of non-split maximal tori is straightforward. Take an element  $u$ of the base field $F$ which is not a square. Then you
get such a torus as the set of matrices
$$
\left(
\begin{array}{cc}
x & uy \newline 
y & x
\end{array}
\right)
$$
where $x^2 -uy^2 =1$. Any maximal non-split torus is conjugate to this torus for some well chosen $u$.
A: 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 $F[\gamma]^\times$.
