Restricted Lie algebra

Remarks

Example: let $A$ be an associative algebra over $F$ with bracket $[x, y] = xy - yx$, and put $x^{[p]} = x^p$. Clause 1 holds because $\operatorname{ad} x = L_x - R_x$, where the left and right multiplications $L_x$ and $R_x$ commute, so $(\operatorname{ad} x)^p = L_x^p - R_x^p = \operatorname{ad}(x^p)$ in characteristic $p$. Clause 2 is clear, and clause 3 is Jacobson's formula for $(x + y)^p$ in an associative algebra. In particular $\mathfrak{gl}_n(F)$ is restricted with the matrix $p$-th power. Non-example: on $\mathfrak{gl}_n(F)$ with $F \ne \F_p$, the identity map $x \mapsto x$ is not a $p$-map, since clause 2 fails for $a \in F$ with $a^p \ne a$. If $\mathfrak{g}$ has trivial centre, there is at most one $p$-map, because clause 1 determines $x^{[p]}$. The algebra of derivations of any algebra over $F$ is restricted with $D^{[p]} = D^p$, since the Leibniz rule gives $D^p(ab) = D^p(a) b + a D^p(b)$ in characteristic $p$.