Let $G$ be a commutative connected algebraic group over $\mathbb{C}$. A theorem of Serre says that there exists an exact sequence $$1\to \mathbb{G}_a^n\times \mathbb{G}_m^m\to G\to A\to 1,$$ where $A$ is an abelian variety. (See here, for example.)

I wonder what are some examples of such groups $G$ where this extension is non-trivial. That is, such that $G\ncong \mathbb{G}_a^n\times \mathbb{G}_m^m\times A$.