Exercise 20, § 1

Let $g$ be a Lie algebra over a ring $K$ such that $pK = \{0\}$ ($p$ prime). A mapping $x \mapsto x^{[p]}$ of $g$ into itself is called a $p$-mapping if it satisfies the relations

for $x, y$ in $g$ and $\lambda \in K$. A Lie $p$-algebra over $K$ is a set with the structure defined by giving it a Lie algebra structure over $K$ and a $p$-mapping. Every associative algebra over $K$ is a Lie $p$-algebra with $[x, y] = xy - yx$ and $x^{[p]} = x^p$ (Exercise 19). A mapping $u$ of a Lie $p$-algebra $g$ into a Lie $p$-algebra $g'$ is called a $p$-homomorphism if $u$ is a Lie algebra homomorphism and $u(x^{[p]}) = (u(x))^{[p]}$. Show that in a Lie $p$-algebra $g$ every $p$-mapping is of the form $x \mapsto x^{[p]} + f(x)$, where $f$ is a mapping of $g$ into its centre, which is semi-linear for the endomorphism $\lambda \mapsto \lambda^p$ of $K$. More generally, if $u$ is a homomorphism of $g$ onto a Lie $p$-algebra $g'$, $u(x^{[p]}) - (u(x))^{[p]}$ belongs to the centre of $g'$.