Question: why are there no non-trivial extensions of $\mathbb{G}_m$ by abelian varieties?

Specifically, let $A$ be an abelian variety over a field $k$. Then it seems to be well known that any extension $$ 0 \to A \to G \to \mathbb{G}_m \to 0$$ in the category of group varieties over $k$ is split. For instance, this is mentioned in [Deligne's Theorie de Hodge I][1] page 426 just before Section 3. Chevalley's structure theorem of algebraic groups certainly implies this.

Question: Is there a simple/direct proof?

Note that there are non-trivial extensions of abelian varieties by tori (semiabelian varieties).

Thanks. [1]: http://mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0425.0430.ocr.pdf

Question: why are there no non-trivial extensions of $\mathbb{G}_m$ by abelian varieties?

Specifically, let $A$ be an abelian variety over a field $k$. Then it seems to be well known that any extension $$ 0 \to A \to G \to \mathbb{G}_m \to 0$$ in the category of group varieties over $k$ is split. For instance, this is mentioned in Deligne's Theorie de Hodge I page 426 just before Section 3. Chevalley's structure theorem of algebraic groups certainly implies this.

Question: Is there a simple/direct proof?

Note that there are non-trivial extensions of abelian varieties by tori (semiabelian varieties).

Thanks.