If $G$ is a connected, reductive, complex group with Borel subgroup $B < G$ and Weyl group $W$, we can write
$$G = \bigsqcup_{w \in W} B w B$$
If $\mathfrak{g}$ is the Lie algebra of $G$, we have an exponential map $\exp : \mathfrak{g} \to G$, so we get a decomposition
$$\mathfrak{g} = \bigsqcup_{w \in W} \exp^{-1}(B w B).$$
Does this accomplish anything? Has it been studied?