  Let $G$ be a simple algebraic group over an algebraically closed field $K$
and let $N = N_G(T)$ be the normalizer of a fixed maximal torus $T\leq G$.
Further, let $U$ be the unipotent radical of a fixed Borel subgroup $B$ that
contains $T$ and let $U^-$ be the unipotent radical of the opposite Borel
subgroup $B^-$. The Bruhat decomposition implies the decomposition $G =
NU^-UN$. The Zariski closed subset $U^-U\subset G$ is isomorphic to the affine
space $A_K^m$ where $m = \dim G -\dim T$ is the number of roots in the
corresponding root system. Here we construct a subgroup $\mathcal{N}\leq
\operatorname{Cr}_m(K)$ that ``acts partially'' on $A_K^m\approx\mathcal{U}$
and we show that there is one-to-one correspondence between the orbits of such
a partial action and the set of double cosets $\{NgN\}$. Here we also calculate
the set $\{g_\alpha\}_{\alpha \in \mathfrak A}\subset \mathcal{U}$ in the
simplest case $G = \operatorname{SL}_2(\mathbb{C})$.
