Let $G$ be a simply-connected, connected Lie group with solvable radical $R$. Then by Levi-Malcev theorem there is a closed semi-simple subgroup $S$ in $G$ so that $G=R\rtimes S$.
I want to show that the condition of simply-connectedness is essential. I found an example that $GL_n(\mathbb C)=\mathbb C^*\cdot SL_n(\mathbb C)$ but $\mathbb C^*\cap SL_n(\mathbb C)\not =\{e\}$. I would like to understand this example, how to prove that the radical is $\mathbb C^*$ and why the intersection is not the identity? Thanks!