The center of an algebraic group when the unipotent radical is abelian

Question

Let $G$ be a connected algebraic complex linear group, such that $G=R_u\cdot S$ where $R_u$ is the unipotent radical and $S$ is the maximal semisimple subgroup.

If $R_u$ is abelian, then is it true that the center $Z$ of $G$ is not trivial?

Answer 1

No. Take $G$ to be any centre-free semi-simple connected complex algebraic group (e.g $G=PGL_n(\mathbb{C})$)

A less obvious counter-example is $G=SL_n(\mathbb{C})\ltimes \mathbb{C}^n$.