If $G$ is an algebraic group with coordinate algebra $A=\mathcal O(G)$, say over a field $k$ of characteristic $p$, then its Lie algebra $\mathfrak g$ can be endowed with the structure of a restricted Lie algebra by defining a $p$-operation $\mathfrak g\to\mathfrak g$ via the inclusion $$ \mathfrak g \to \mathrm{Der}_k(A,A), $$ and using $p$-th power maps: the $p$-operation is simply the $p$-th power in $\mathrm{Hom}_k(A,A)$ which, as it turns out, sends left-invariant derivations to left-invariant derivations, i.e. maps $\mathfrak g$ into itself.
However, $\mathfrak g$ can also be thought of as the vector space $\mathrm{Der}_k(A,k_1)$, where $k_1$ is $k$ as an $A$-module through the identity $1\in G$ (which is a map $A\to k$ with kernel $\mathfrak m$) or as a vector space $\mathrm{Hom}_k(\mathfrak{m}/\mathfrak{m}^2,k)$.
My question: is it possible to define the $p$-operation directly on $\mathrm{Der}_k(A,k_1)$? In fact, every alternative way to introduce and study the $p$-operation would be kindly appreciated especially those most easily expressed in the scheme-theoretic language for algebraic groups.
A remark - we can also define the $p$-operation directly on the Lie algebra as follows (Jacobson proved that this is equivalent with being the restricted Lie algebra of an algebraic group).
A restricted $p$-Lie algebra is a Lie algebra $L$ over a field $K$ together with a $p$-map $[p]\colon \mathfrak{g} \to \mathfrak{g},\;x \mapsto x^{[p]}$, such that
$(1)$ ${\rm ad} \:x^{[p]}=(ad \:x)^p \quad \forall \;x \in \mathfrak{g}$
$(2)$ $(\alpha x)^{[p]}=\alpha^p x^{[p]} \quad \forall \;x \in \mathfrak{g},\;\alpha \in K$
$(3)$ $ad \:(x+y)^{[p]}=x^{[p]}+y^{[p]}+\sum_{i=1}^{p-1} s_i(x,y)$, where $i\:s_i(x,y)$ is the coefficient of $X^{i-1}$ in the polynomial $(ad \,(xX+y)) ^{p-1}(x)$.