Let $k$ be a number field and let $G$ be a connected semisimple, simply connected algebraic group defined over $k$. Let $k'$ be a finite Galois extension over which $G$ splits. By the Chebotarev Density Theorem, there are an infinite number of places $v$ of $k$ such that $v$ splits completely in $k'$. The paper I'm reading states without proof that over all such $v$, $G \times_k k_v$ is split. Why is this true?
Is the simply connected or even semisimple hypothesis necessary?