Closed subgroup schemes are detected on all algebra-valued points

If $H$ is a closed subgroup scheme, its structure morphisms give a group law on $H(R)$ for every $R$ and its inclusion in $G(R)$ preserves the three operations by [F1]. Thus $H(R)$ is a subgroup. Conversely suppose every $H(R)$ is a subgroup. Taking $R=k$ shows that the identity $e_G\in G(k)$ has a factor $e_H:\mathrm{Spec}k\to H$.

Cover $H{\times }_{k}H$ by affine opens $U=\mathrm{Spec}R$. The restrictions of the two projections to $U$ are points $a,b\in H(R)$. By hypothesis their product in $G(R)$ lies in $H(R)$, so $m_G\circ (j\times j){\mathrm{\mid }}_U$ factors through $H$. The factors agree on every overlap by uniqueness through the closed immersion and hence glue to $m_H:H{\times }_{k}H\to H$. Similarly, on every affine open $\mathrm{Spec}R\subset H$, its inclusion is a point of $H(R)$, whose inverse in $G(R)$ belongs to $H(R)$. These factors glue uniquely to $i_H:H\to H$. This proves all three factorization assertions using universal affine points, rather than only field-valued points.

Compose the associativity, identity and inverse identities for these factors with $j$. They become precisely the corresponding identities in $G$ by construction. Since $j$ is a monomorphism, the identities hold in $H$. The closed scheme $H$ is finite type over $k$: a finite affine cover $\mathrm{Spec}{A}_i$ of the finite-type $G$ pulls back by [F2] to $\mathrm{Spec}(A_i\mathrm{/}{I}_i)$, a finite affine cover with finitely generated $k$-algebras. Thus $H$ is a group scheme of finite type and $j$ a group-scheme morphism by [F1]. Uniqueness of every factor proves uniqueness of its group law. The converse for the equivalent factorization criterion follows from exactly the same transfer of identities. AC is inherited through the affine quotient presentation in [F2], as recorded in [F3].