Exercise 5, § 2

Suppose that $\mathbf{K}$ is of characteristic $p > 0$ ($p$ prime). Let $g$ be a Lie algebra over $\mathbf{K}$ with a basis, canonically identified with a submodule of its enveloping algebra $U$. For an endomorphism $\phi$ of the $\mathbf{K}$-module $g$ to be a $p$-mapping, it is necessary and sufficient that $x \mapsto x^p - \phi(x)$ be a semi-linear mapping (for λ ↦ λ^p) of g into the centre of U. Deduce that if (b_λ) is a basis of g, for there to exist a p-mapping on g, it is necessary and sufficient that, for all λ, there exist c_λ ∈ g such that (ad b_λ)^p = ad c_λ; if this is so, there then exists one and only one p-mapping x ↦ x^{[p]} such that b_λ^{(p)} = c_λ for all λ.