Let $G$ be a commutative 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][1], for example.)

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


  [1]: https://icerm.brown.edu/materials/Slides/sp-s12-w3/On_Mordell-Lang_in_algebraic_groups_of_unipotent_rank_1_]_Paul_Vojta,_University_of_California,_Berkeley.pdf