Exercise 7, § 2

Let g be a Lie p-algebra over a ring K such that pK = {0} (p prime). A derivation D of g is called a p-derivation if D(x^p) = (ad x)^{p−1}. Dx for all x ∈ g. Every inner derivation is a p-derivation.

(a) If L is an associative K-algebra, every derivation of L is a p-derivation when L is considered as a Lie p-algebra. (Use formula (2) of Exercise 19 of § 1.)

(b) Suppose that g has a basis. For a derivation of g to be a p-derivation, it is necessary and sufficient that it can be extended to a derivation of the restricted enveloping algebra of g. Deduce that the p-derivations of g form a Lie p-subalgebra of the p-algebra of derivations of g.

(c) If D is a p-derivation of g, the kernel of D is a p-subalgebra of g.

(d) For every derivation D of g, D(x^p) − (ad x)^{p−1}. Dx belongs to the centre of g for all x ∈ g (use formula (2) of Exercise 19 of § 1, applied to L = $\mathscr{L}(g)$).