Let $G$ be a connected reductive group (although I don't think this is relevant here), $B$ a Borel subgroup containing a maximal torus $T$ and $U$ the associated unipotent radical ($U^-$ the opposite unipotent subgroup). Finally, let $v \in W$ and $\dot{v}$ a lift of $v$ in $N_G(T)$.

How do you prove that there is an isomophism of varieties $$ (U\dot{v} \cap \dot{v}U^-) \times U \to U\dot{v}U, $$ which is the multiplication $(x,y) \mapsto xy$ ?

Thank you!