The setup:
Let $K$ be a field which is not of characteristic 2 and which contains at least seven elements. Let $N$ be a normal subgroup of $\operatorname{SL}_2(K)$ which contains a matrix $A \neq \pm I$.
The problem:
(a) Show that $N$ contains an upper triangular matrix other than $\pm I$.
(b) Show that $N$ contains a unit upper triangular matrix other than $\pm I$.
Now the problem goes on, and the end goal is to show that there are infinitely many nonabelian simple groups. I've got part (c) through (f), but I simply can't get anywhere with these two.
There is a hint attached to part (a), suggesting we conjugate to get a 0, then compute a commutator with a diagonal matrix to place the 0.
Conjugation isn't exactly the most elegant operation here. For any $\begin{pmatrix} a&b\\c&d \end{pmatrix}$ in $N$, and any $\begin{pmatrix} r&s\\t&u\end{pmatrix}$ in $\operatorname{SL}_2(K)$, we have $ \begin{pmatrix} u&-s\\-t&r\end{pmatrix}\begin{pmatrix} a&b\\c&d\end{pmatrix}\begin{pmatrix} r&s\\t&u\end{pmatrix} = \begin{pmatrix} u(ar+bt)-s(cr+dt)&u(as+bu)-s(cs+du)\\r(cr+dt)-t(ar+bt)&r(cs+du)-t(as+bu)\end{pmatrix},$ and none of those look particularly easy to force to be 0. (I can get the bottom left to $-b$ or $-c$, but those haven't been helpful on their own.)
I don't know what to do for the commutator yet, but to be fair, I don't know where my 0 is supposed to be, so that will hopefully resolve with the issue above.
I'm hoping my problems with part (b) also stem from my lack of progress with part (a), so any hints would be greatly appreciated. Thanks!