I am interested in the following problem : given a finite field $F$ and two unipotent elements $g\_1,g\_2\in\mathrm{SL}\_2(F)$ which do not commute, what can we say about the subgroup they generate? More specifically, is there an additional condition under which it is known that they generate all of $\mathrm{SL}\_2(F)$? 


For $F=\mathbf F\_p$ it is easily seen that this is always the case. For $F=\mathbf{F}\_{p^2}$ I convinced myself a while ago (using lenghty computations of products of $g\_1,g\_2$) that they generate either the whole of $\mathrm{SL}\_2(F)$ or a subgroup conjugated to $\mathrm{SL}\_2(\mathbf{F}\_p)$. I would be glad to see a more enlightening proof of this fact, and even more if it pointed at generalizations to other algebraic groups than $\mathrm{SL}\_2$. 



Subgroups of $SL_2(F)$ generated by unipotent elements
You should look up Dickson's Theorem. It deals pretty comprehensively with the two dimensional case when $F$ is finite, at least in odd characteristic. The proof may be found in Gorenstein's 1968 book "Finite Groups". I don't remember the statement precisely, but I think it's that $\langle \left( \begin{array}{clcr} 1 & 0\\1 &1 \end{array} \right),\left( \begin{array}{clcr} 1 & \lambda\\0 &1 \end{array} \right) \rangle $ is all of ${\rm SL}(2,K)$, where $K$ is the subfield of $F$ generated by $\lambda \neq 0,$ except when $|F| =9,$ in which case ${\rm SL}(2,5)$ may occur.


There is also the work by J.G. Thompson on quadratic pairs, which may be seen as a massive generalization of this result. Going back further historically than that, there is the Hall-Higman theorem, and more recently there has been extensive work on so-called $2F$-modules, which is relevant to the classification of finite simple groups.


