Let $g \in GL\_n(\mathbb{F}\_q)$. Is it true that we can always write $g = u\_1lu\_2$, where $u\_1$ and $u\_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices be unipotent.


This is equivalent to being able to write $g=u\_1lu\_2d$, where $u\_1$ and $u\_2$ are unipotent upper-triangular, $l$ is unipotent lower-triangular, and $d$ is a diagonal matrix (see Geoff's comment below).


Remark: It suffices to show that this is true for $g$ a permutation matrix; the Bruhat decomposition will then guarantee that this will be true for arbitrary $g$. In particular, the statement is true for $n=2$ as
$$
 \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.
$$



ULU Decomposition of a matrix
The keyword for this is "Gauss decomposition". It states that for a ring R of stable rank 1 one can indeed write any element g of GL(n,R) as a product of three upper and lower triangular matrices (not unipotent, though). So it holds not only for finite fields, but for any field and, more generally, semilocal ring, along with many other examples such as the ring of all algebraic integers. And for GL one doesn't even require R to be commutative.


I'm having trouble tracing the first appearence of this in the literature (but probably it goes back to the work of Bass), so here are two links:


* A [paper](http://www.theoryofgroups.ir/article_467_59.html), where this is proved for any [elementary] Chevalley groups (not extended, but the idea works for them too), and the ring is assumed to be commutative. It also contains an overview of the previous results on the topic;
* Another [paper](http://www.sciencedirect.com/science/article/pii/002437959090270M), where this is proved for $GL\_n$ over possibly non-commutative rings of stable rank 1.


If you ask the matrices to be unipotent, then, first, you can only get matrices from $SL\_n$, and second, three is not enough, the torus is the obstacle. But if you take four upper and lower unipotent triangular matrices, you can do it (still in $SL\_n$). For this the keyword is "[unitriangular factorisation](http://arxiv.org/abs/1107.5414)".


