I am trying to learn arithmetic groups, from a dynamical point of view. These questions (maybe silly) come to my mind, but I do not know the answer.

How does $SL_2(\mathbb Z[\sqrt 2])$ embed into $SL(2,\mathbb R)\times SL(2,\mathbb R)$? (The former is an irreducible lattice in the latter.)

What are the centralizers of $SL_2(\mathbb Z[\sqrt 2])$ in $SL(4,\mathbb R)$? Trivial?

===========================================================================

I notice that the above questions are rather silly in some sense. Let me include another two (maybe also silly) questions.

Why $SL_2(\mathbb Z[\sqrt 2])$ is a lattice in $SL(2,\mathbb R)\times SL(2,\mathbb R)$? Why not in $SL(2,\mathbb R)$ as $SL(2,\mathbb Z)$?

How does $SL_2(\mathbb Z[\sqrt 2])$ embed into $SL(4,\mathbb Z)$? So acts on $\mathbb T^4$.

=========================================================================== Add on Question 4.

The embedding can be constructed as follows. For any $A+B\sqrt 2\in SL_2(\mathbb Z[\sqrt 2])$ with $A,B$ integer matrices. Let $\tau:A+B\sqrt 2\mapsto \begin{bmatrix} A & 2B\\ B&A\end{bmatrix}\in SL(4,\mathbb Z)$. One can check $\tau$ is indeed a group homomorphism.

===========================================================================

Answers, references and comments are highly appreciated.

notreduced to scalars). $\endgroup$ – YCor Jun 5 '18 at 15:29