Multiple rational normal forms in Lie theory

We study the decomposition of a generic element $g\in G$ of a connected reductive complex algebraic group $G$ in the form $g=N(g)B(g)\bar{u}N(g{)}^{-1}$ where $N:G⇢{\mathcal{N}}_{-}$ and $B:G⇢{\mathcal{B}}_{+}$ are rational maps onto a unipotent subgroup ${\mathcal{N}}_{-}$ and a Borel subgroup ${\mathcal{B}}_{+}$ opposite to ${\mathcal{N}}_{-}$, and $\bar{u}$ is a representative of a Weyl group element $u$. We introduce a class of rational Weyl group elements that give rise to such decompositions, and study their various properties.