Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ is trivial. I would like to know an example of a simple algebraic group such that the base extension $G_{\overline{k}}$ of $G$ to the algebraic closure $\overline{k}$ of $k$ is not simple anymore.
If $G$ is connected and non-commutative then also $G_{\overline{k}}$ is connected and non-commutative. So the problem is really about normal subgroups of $G_{\overline{k}}$ not being defined over $k$.