Here is the general Bruhat decomposition (which I have seen in various paper but never with a proof or a complete reference).

Let $G$ be a split reductive group, $T$ a split maximal torus and $B$ a Borel subgroup of $G$.

Let $R^+ \subset R$ be the positive roots corresponding to $B$ and $S \subset R^+$ the simple roots of $R^+$. Let $I \subset S$ and $P_I$ the standard parabolic subgroup of $G$ corresponding to $I$.

Finally let $W$ be the Weyl group of $(G,T)$ and $W_I$ the subgroup of $W$ generated by the reflections $(s_\alpha)_{\alpha \in I}$.

Then the general Bruhat decomposition is $$G = \coprod_{W_I \backslash W / W_I} P_I w P_I$$ and $P_I \backslash P_I w P_I$ is an affine variety of dimension $\ell(w)$ where $w$ is of minimal length in the double coset $P_I w P_I$.

My question is : is there a good choice of representatives for $P_I \backslash P_I w P_I$ ? More precisely, I am looking for an analogue of the following bijection (in the case $P=B$ Borel) : $$B \times \lbrace w \rbrace \times U_{w^{-1}} \overset{\sim}{\longrightarrow} BwB$$ where $U$ is the unipotent radical of $B$, $U^-$ its opposite and $U_{w^{-1}}$ is the subgroup $(w^{-1}U^-w) \cap U$. What subgroup of $P_I$ would replace $U_{w^{-1}}$ ?

Also what reference exists for all this ?

Thanks in advance.

Edit : in this course of Casselman I found the following isomorphism of variety (see on top of page 12)

$$P_I \times \lbrace w \rbrace \times \prod_{\alpha \in R^+ \backslash R_I^+ ~|~ w^{-1} \alpha \notin R^+ \backslash R_I^+} N_\alpha \overset{\sim}{\longrightarrow} P_IwP_I$$

with $w \in W$ of minimal length in $W_I \backslash W / W_I$. However this seems not to work with $\mathrm{GL_3}$ : we note $S = \lbrace \alpha, \beta \rbrace$ ; if $I= \lbrace \alpha \rbrace$, $P_I = \left( \begin{smallmatrix} * & * & * \newline * & * & * \newline & & * \end{smallmatrix} \right)$ ; with $w = s_\beta$ the above product is on the set $\lbrace \beta, \alpha + \beta \rbrace$, so the isomorphism should be $P_I s_\beta P_I \cong P_I \times \lbrace s_\beta \rbrace \times \left( \begin{smallmatrix} 1 & 0 & * \newline 0 & 1 & * \newline 0 & 0 & 1 \end{smallmatrix} \right)$, which is false (the element $s_\beta \left( \begin{smallmatrix} 1 & 0 & 0 \newline 1 & 1 & 0 \newline 0 & 0 & 1 \end{smallmatrix} \right)$ is in the left side, not in the right side)...