In what follows, I will refer to prop. 0.8 in SGA 3 Exp. III which can be found for example at the link (https://webusers.imj-prg.fr/~patrick.polo/SGA3/). I will quickly introduce the notation without too many details to give the context, sorry for any imprecisions.
Let $S$ be a base scheme and let $X$ be a $S$-scheme. Let $P : X \times X \rightarrow X$ a $S$-morphism of schemes. Let $I ,J$ be (quasi-coherent) ideal of $S$ such that $J^2=0$ and $J \subset I$. Denote respectively $S_0$, $S_J$ the corresponding base schemes and $X_0$, $X_J$ the attached subschemes of $X$. Assume $X_J$ is a $S_J$-group scheme and that the restriction $P_J$ is its group law. Let $L_X$ be the abelian group scheme given essentially by $\operatorname{Hom}_{\cdot} (\Omega^1_{X_0 / S_0} , J \otimes \cdot)$. After specializing everything on points say over $S'$, one should have that for all derivations $m, m' \in L_X (S')$ and for all points $x, x' \in X(S')$:
$$P(m\cdot x, m' \cdot x')=(m \times \operatorname{Ad}(g) m')\cdot P(x, x')$$
where $\cdot$ denotes the action of $L_X$ on $X$, $\times$ denotes the group law on $L_X$ and $\operatorname{Ad}$ denotes the adjoint representation.
My confusion arises from the fact that on the RHS it looks like I might have $m$ acting on $x'$ and $m'$ acting on $x$ which should not happen on the LHS.
Let's consider a simple example where the adjoint representation is trivial. Take $R$ local artin ring of characteristic $p$ and consider $\alpha_p$ over $R$. We have $J=(\pi)$ and $I=\mathfrak{m}_R$. Take in $R$ an element $\pi$ such that $\pi \in \operatorname{Ann} (\mathfrak{m}_R)$ (where $\mathfrak{m}_R$ is the maximal ideal of $R$). Now we should have that:
$$m\cdot x + m'\cdot x' = (m+m') \cdot (x+x')$$
but $m\cdot x$ should be equal to $x + D_m (x)$ where $D_m$ is the derivation attached to $m$, this is exactly $m$ evaluated in $x$. So we should conclude that
$$x+m(x)+x' +m' (x') = x+x' + m(x) + m(x') +m' (x) + m' (x') $$
which looks clearly false.
