$G$ satisfies the sheaf property for the Zariski topology,
there exists a subfunctor $F \subset G$ such that
$F$ is representable,
$F \subset G$ is representable by open immersion,
for every field extension $K$ of $k$ and $g \in G(K)$ there exists a $g' \in G(k)$ such that $g'g \in F(K)$.
Then $G$ is representable by a group scheme over $k$.