as a $\k$-group functor, the Lie algebra $\g$ of $\G$ is defined by $\g(\R):={\rm Ker}(\G(\rp_\R))$ where for any $\k$-algebra $\R$ we put $\rp_\R:\R[\epsilon]\rightarrow \R$, $a+\epsilon b\mapsto a$. Denote by $e:\k[\G]\rightarrow\k$ the counit (the identity element in $\G(\k)$) and put $\I={\rm Ker}(e)$. Then for any $\k$-algebra $\R$, we have [@milne §10.19]

$$\g(\R)\cong{\rm Hom}_\k(\I/\I^2,\R).$$ From this we see that $\g(\R)$ is an Abelian group in fact an $\R$-module. We have that the canonical morphism

()\_=[Hom]{}\_(/\^2,)\_\_(/\^2,)=() is an isomorphism [@Bour §2, Proposition 2].

By definition $\g(\R)$ is a normal subgroup of $\G(\R[\epsilon])$ and so the latter acts on $\g(\R)$ by conjugation. As $\g(\R)$ is Abelian this action factors through an action of $\G(\R)=\G(\R[\epsilon])/\g(\R)$ on $\g(\R)$. This is the so-called adjoint action of $\G$ on its Lie algebra, denoted :()(()).Using the section $i_\R:\R\rightarrow\R[\epsilon]$, $a\mapsto a$ of $\rp_\R$ we see that the exact sequence $$0\rightarrow \g(\R)\rightarrow\G(\R[\epsilon])\rightarrow \G(\R)\rightarrow 1$$ splits and so

$$\G(\R[\epsilon])\simeq \g(\R)\rtimes\G(\R).$$

From now on we assume that $\V$ is a finite-dimensional $\k$-vector space which we regard as a $\k$-group functor

$$\V(\R):=\V\otimes_\k\R,$$ for any $\k$-algebra $\R$. Denote by $\GL(\V)$ the $\k$-group scheme with $\k$-group functor $$\R\mapsto {\rm End}_\R(\V(\R))^\times.$$

Here we collect some simple observations how we can work explicitly with $\GL(\V)(\R[\epsilon])$.

\(i) We can 