The adjoint representation of an affine group scheme

Conjugation preserves the kernel. For $x\in G(R)$ define $c_x:G(R[\epsilon ])\to G(R[\epsilon ])$ by $c_x(y)=xy{x}^{-1}$, using the group structure of [F3]; it is an automorphism with inverse $c_{x^{-1}}$, and it is induced by the automorphism of the functor $G$ given by conjugation with the image of $x$ under $G(R)\to G(R[\epsilon ])$. Since the reduction $\rho :G(R[\epsilon ])\to G(R)$ is a homomorphism of groups and the image of $x$ in $G(R[\epsilon ])$ reduces to $x$, one has $\rho (c_x(y))=x\rho (y)x^{-1}$; hence $c_x$ maps $\mathrm{Lie}(G)(R)=\ker \rho$ into itself, and so does $c_{x^{-1}}$. Define $\mathrm{Ad}(x):=c_x{\mathrm{\mid }}_{\mathrm{Lie}(G)(R)}$.

The maps $\mathrm{Ad}(x)$ and the map $x\mapsto \mathrm{Ad}(x)$. Each $\mathrm{Ad}(x)$ is a group automorphism of $\mathrm{Lie}(G)(R)$ by step 1.1, hence is additive because the group law there is addition by [F1]; it is $R$-linear because the scalar action of $c\in R$ on $\mathrm{Lie}(G)(R)$ is induced by the algebra endomorphism $\epsilon \mapsto c\epsilon$ of $R[\epsilon ]$, which commutes with the conjugation $c_x$ since the image of $x$ in $G(R[\epsilon ])$ is fixed by that endomorphism. Moreover $\mathrm{Ad}(xy)=\mathrm{Ad}(x)\circ \mathrm{Ad}(y)$ and $\mathrm{Ad}(e)=\mathrm{id}$ because $c_{xy}=c_x\circ c_y$, and the construction is natural in $R$ because both the group structures and the reduction maps are. Thus $x\mapsto \mathrm{Ad}(x)$ is a natural homomorphism from the group-valued functor $h_G$ to the functor $R\mapsto {\mathrm{Aut}}_R(\mathfrak{g}{\otimes }_{k}R)$, which under the identification of [F1] is exactly the group of $R$-linear automorphisms of $\mathrm{Lie}(G)(R)$.

Clause (b). For $X\in \mathfrak{g}{\otimes }_{k}R$ the element $e^{\epsilon X}$ of $\mathrm{Lie}(G)(R)$ is fixed by the identification of [F2], and by definition $\mathrm{Ad}(x)$ is the restriction of conjugation by $x$; hence $x{e}^{\epsilon X}{x}^{-1}=e^{\epsilon \mathrm{Ad}(x)X}$ in $G(R[\epsilon ])$.

Clause (c), naturality. Let $f:G\to H$ be a morphism of affine group schemes of finite type over $k$ and let $x\in G(R)$, $X\in \mathfrak{g}{\otimes }_{k}R$. Applying the group homomorphism $f_{R[\epsilon ]}$ to the identity of clause (b) for $G$ gives $f(x)f(e^{\epsilon X})f(x{)}^{-1}=f(e^{\epsilon {\mathrm{Ad}}_G(x)X})$; by the functoriality of $f$ on points and the exponential identity of [F1] this reads $e^{\epsilon {\mathrm{Ad}}_H(f(x))\mathrm{Lie}(f)X}=e^{\epsilon \mathrm{Lie}(f){\mathrm{Ad}}_G(x)X}$, and the exponential correspondence $X\mapsto e^{\epsilon X}$ is injective, so ${\mathrm{Ad}}_H(f(x))\circ \mathrm{Lie}(f)=\mathrm{Lie}(f)\circ {\mathrm{Ad}}_G(x)$.

The adjoint representation is a morphism. If $\mathfrak{g}=0$, its automorphism functor is the one-element functor, represented by the trivial group $\mathrm{Spec}k$, and $\mathrm{Ad}$ is its unique morphism. Otherwise, by [F1] the Lie algebra $\mathfrak{g}$ is finite-dimensional, so by [F5] it has a finite $k$-basis and the functor $R\mapsto {\mathrm{Aut}}_R(\mathfrak{g}{\otimes }_{k}R)$ is naturally identified with ${\mathrm{GL}}_n$ for $n={\dim }_k\mathfrak{g}$; by The general linear group scheme and its coordinate ring this functor is the functor of points of the affine group scheme ${\mathrm{GL}}_{\mathfrak{g}}={\mathrm{GL}}_n$. The natural transformation of step 2.1 is therefore a natural transformation $h_G\to h_{{\mathrm{GL}}_{\mathfrak{g}}}$, and by [F4] it is induced by a morphism of $k$-schemes $\mathrm{Ad}:G\to {\mathrm{GL}}_{\mathfrak{g}}$, which is a morphism of group schemes because the transformation is a natural homomorphism of group-valued functors.

Conclusion. Steps 1.1, 2.1, 2.2, 3.1 and 3.2 prove (a), (b) and (c): conjugation restricts to an $R$-linear automorphism of the Lie algebra, the assignment is a natural group homomorphism and hence defines the morphism $\mathrm{Ad}:G\to {\mathrm{GL}}_{\mathfrak{g}}$ of $k$-group schemes, the identity of (b) is the definition of the restriction, and (c) is the differentiated naturality. The conjugation construction and finite basis selection are choice-free; the finite-type assertion for ${\mathrm{GL}}_{\mathfrak{g}}$ inherits Choice from The general linear group scheme and its coordinate ring.