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 $$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. $$
