Let $G$ be a connected (not necessarily simply-connected) Lie group (which is not necessarily linear, not necessarily algebraic), and $\mathfrak{g}$ be its Lie algebra.
It is known that the Levi decomposition of its Lie algebra $\mathfrak{g}$ splits, i.e. $$ \mathfrak{g} \cong \mathfrak{r} \rtimes \mathfrak{s} $$ where $\mathfrak{r}$ is a radical of $\mathfrak{g}$ (maximal solvable ideal), and $\mathfrak{s}$ a semisimple Lie subalgebra of $\mathfrak{g}$.
- Does this extend to a parallel statement in $G$? That means, is it true that $$ G \cong R \rtimes S $$ where $R$ is the radical (maximal connected normal solvable Lie subgroup) of $G$ and $S$ some semisimple Lie subgroup of $G$? If yes, how can we see it directly from the Lie algebra statement? (My main problem is, I'm not sure how $\exp$ interacts with $\rtimes$.)
(I know for sure that, with $R$ being the radical of $G$, $G/R$ is semisimple. The main issue lies in the split property of $1 \to R \to G \to G/R \to 1$).
This must be a very basic question, but I'm not familiar enough with the path between Lie groups and Lie algebra to trust my intuition...