Let $G$ be a Lie group, there is a natural inclusion of $G$ in Diff($G$), by left traslation.
Is true that there is a splitting Diff($G$)$\cong$ $G$ $\times$ Diff($G,e$)? Where Diff($G,e$) is the subgroup of Diff($G$) that fixes the identity element of the group.
This question comes from the example 1 of https://en.wikipedia.org/wiki/Diffeomorphism#Diffeomorphism_group