ag.algebraic geometry - Is the field of invariants $k(V)^G$ purely transcendental over $k$?

Reference: http://www.math.u-psud.fr/~colliot/mumbai04.pdf

Proposition 4.3. on page 18 in the above reference reads as follows:
Assume $k = overline{k}$. If $V$ is a finite dimensional vector space over $k$ and $G subset GL(V)$ is an (abstract) abelian group consisting of semisimple elements, then $k(V)^G$ is pure.

I would like to find an abelian group $G subset GL(V)$ such that $k(V)^G$ is not pure (if it exists it would need to be infinite due to Fischer's theorem, and not a connected solvable group according to Proposition 4.4).

Thanks.