This lemma generalizes Lemma \ref{lemma-group-space-scheme-locally-finite-type-over-k} (which covers all cases one cares about in practice). The proof is very similar to the proof of Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines} used in the proof of Lemma \ref{lemma-group-space-scheme-locally-finite-type-over-k} and we encourage the reader to read that proof first. \medskip\noindent By Lemma \ref{lemma-group-space-scheme-over-kbar} the base change $G_{\overline{k}}$ is a scheme. Let $K/k$ be a purely transcendental extension of very large transcendence degree. By Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-scheme-after-purely-transcendental-base-change} it suffices to show that $G_K$ is a scheme. Let $K^{perf}$ be the perfect closure of $K$. By Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change} it suffices to show that $G_{K^{perf}}$ is a scheme. Let $K \subset K^{perf} \subset \overline{K}$ be the algebraic closure of $K$. We may choose an embedding $\overline{k} \to \overline{K}$ over $k$, so that $G_{\overline{K}}$ is the base change of the scheme $G_{\overline{k}}$ by $\overline{k} \to \overline{K}$. By Varieties, Lemma \ref{varieties-lemma-make-Jacobson} we see that $G_{\overline{K}}$ is a Jacobson scheme all of whose closed points have residue field $\overline{K}$. \medskip\noindent Since $G_{\overline{K}} \to G_{K^{perf}}$ is surjective, it suffices to show that the image $g \in |G_{K^{perf}}|$ of an arbitrary closed point of $G_{\overline{K}}$ is in the schematic locus of $G_K$. In particular, we may represent $g$ by a morphism $g : \Spec(L) \to G_{K^{perf}}$ where $L/K^{perf}$ is separable algebraic (for example we can take $L = \overline{K}$). Thus the scheme \begin{align*} T & = \Spec(L) \times_{G_{K^{perf}}} G_{\overline{K}} \\ & = \Spec(L) \times_{\Spec(K^{perf})} \Spec(\overline{K}) \\ & = \Spec(L \otimes_{K^{perf}} \overline{K}) \end{align*} is the spectrum of a $\overline{K}$-algebra which is a filtered colimit of algebras which are finite products of copies of $\overline{K}$. Thus by Groupoids, Lemma \ref{groupoids-lemma-compact-set-in-affine} we can find an affine open $W \subset G_{\overline{K}}$ containing the image of $g_{\overline{K}} : T \to G_{\overline{K}}$. \medskip\noindent Choose a quasi-compact open $V \subset G_{K^{perf}}$ containing the image of $W$. By Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-when-scheme-after-base-change} we see that $V_{K'}$ is a scheme for some finite extension $K'/K^{perf}$. After enlarging $K'$ we may assume that there exists an affine open $U' \subset V_{K'} \subset G_{K'}$ whose base change to $\overline{K}$ recovers $W$ (use that $V_{\overline{K}}$ is the limit of the schemes $V_{K''}$ for $K' \subset K'' \subset \overline{K}$ finite and use Limits, Lemmas \ref{limits-lemma-descend-opens} and \ref{limits-lemma-limit-affine}). We may assume that $K'/K^{perf}$ is a Galois extension (take the normal closure Fields, Lemma \ref{fields-lemma-normal-closure} and use that $K^{perf}$ is perfect). Set $H = \text{Gal}(K'/K^{perf})$. By construction the $H$-invariant closed subscheme $\Spec(L) \times_{G_{K^{perf}}} G_{K'}$ is contained in $U'$. By Spaces over Fields, Lemmas \ref{spaces-over-fields-lemma-base-change-by-Galois} and \ref{spaces-over-fields-lemma-when-quotient-scheme-at-point} we conclude.