I am trying to understand Bruhat decomposition in practique, and more generally matrices decompositions. Let me take the example of $G=GL(3)$, and introduce $B$ the subgroup of upper triangular matrices, $N$ of unipotent upper triangular matrices, and $D$ of diagonal ones. The Bruhat decomposition is $$(1) \qquad G = \bigcup_w BwB$$
where elements $w$ are running through the permutation group in $GL(3)$. I would like to write such a decomposition not with $B$,but with $N$, that is to say compute $N \backslash G / N$. For this I tried to do so for every classes appearing in $(1)$, for instance the trivial one gives
$$B1B = B = NDN$$
However for every other permutation matrices I am lost: should I blindly try, like with general coefficients and doing computations explicitly, or is there a more systematic method?
I think you can start with $B = ND = DN$, to get
$$G = \bigcup_{w} N DwD N$$
The next question is when these classes are distinct. From the Bruhat decomposition, we know that $NDwDN = NDw'DN$ if and only if $w = w'$. And since we have $DwD = Dw = w D$, we can write
$$G = \bigsqcup_{w} \bigcup_{d} N w d N$$
Finally, we prove that $NwdN = Nwd'N$ if and only if $d = d'$ (if $n_1 wd n_2 = wd'$, then $w^{-1}n_1w = d'n_2^{-1}d^{-1}$; the right-hand side is un upper triangular matrix with elements of $d'd^{-1}$ on the diagonal, while the left-hand side is unipotent so we get $d = d'$). In conclusion, we have
$$G = \bigsqcup_{w} \bigsqcup_{d} N w d N$$
As with the Bruhat decomposition, we can interpret this decomposition with the Gauss-Jordan elimination.