The centre is the stable kernel of conjugation on local jets

Put $R={\mathcal{O}}_{G,e}$ and $X_n=\mathrm{Spec}(R\mathrm{/}{\mathfrak{m}}_e^{n+1})$. These are finite subschemes of $G$: on an affine neighbourhood of $e$, localization induces the same quotient by the corresponding maximal-ideal power. Their rings are finite dimensional because $R$ is Noetherian with residue field $k$. Conjugation fixes $e$ scheme theoretically, hence preserves its ideal and every power; it therefore acts on each $X_n$. Pullback by inverse conjugation gives linear automorphisms of $R\mathrm{/}{\mathfrak{m}}_e^{n+1}$ with regular matrix entries, defining the jet representations. Their closed scheme kernels $H_n$ descend with $n$ and stabilize to $H$, since their ideal sheaves ascend on the Noetherian scheme $G$. Every central point acts trivially on all jets, so $Z(G)\subset H$ as functors.

Let $a:H\times G\to G$ be conjugation and $p$ projection. The group $G$ is separated: its identity is a closed rational point, and its diagonal is the inverse image of that point under $(x,y)\mapsto x^{-1}y$. Thus the equalizer of $a,p$ is closed, with ideal sheaf $J$. For an affine chart $\mathrm{Spec}C\subset H$ and an affine neighbourhood $U=\mathrm{Spec}B$ of $e$, every section $z\in J(C{\otimes }_{k}B)$ vanishes in $C{\otimes }_k(R\mathrm{/}{\mathfrak{m}}_e^{n+1})$ for all $n$, because $H$ acts trivially on all jets. Write $z={\sum }_i{c}_i\otimes b_i$ with the finitely many $c_i$ linearly independent over $k$. Finite coefficient contractions show $b_i\in {\mathfrak{m}}_e^{n+1}R$ for every $n$. By [F1] all these coefficients vanish in $R$. Since $G$ is integral, $B\hookrightarrow R$, so $z=0$. Hence $a=p$ on $H\times U$.

The nonempty open $U\subset G$ is schematically dense after every $k$-algebra base change. Indeed for any nonempty affine open $W\subset G$, choose a nonempty principal open $D(b)\subset W\cap U$; integrality makes $\mathcal{O}(W)\hookrightarrow \mathcal{O}(W{)}_b$ injective, and tensoring over $k$ with $C$ preserves injectivity. Consequently a section of $J$ on $\mathrm{Spec}C\times W$ vanishing on $\mathrm{Spec}C\times U$ is zero. Step 2.1 therefore gives $J=0$ on all of $H\times G$. Thus conjugation by $H$ is the identity after arbitrary base change, which says $H\subset Z(G)$. Together with step 1.1 this proves equality scheme theoretically and represents the centre by the stable closed kernel $H$. AC is inherited from [F1]; no faithfulness of conjugation is assumed.