Exercise 23, § 4

Let $g$ be a Lie $p$-algebra over a perfect field $K$ of characteristic $p > 0$. $g$ is called $p$-unipotent if for all $x \in g$ there exists $m$ such that $x^{p^m} = 0$.

(a) Show that every $p$-unipotent Lie $p$-algebra is nilpotent.

(b) Suppose that $g$ is nilpotent and let $h$ denote the $p$-core of the centre of $g$ ($§ 1$, Exercise 23 (a)). Show that $g/h$ is $p$-unipotent.

(c) Let $g$ be the nilpotent Lie $p$-algebra with a basis of three elements $e_1, e_2, e_3$ such that $[e_1, e_2] = [e_1, e_3] = 0,\ [e_2, e_3] = e_1,\ e_1^p = e_1,\ e_2^p = e_3^p = 0$. Then $h = K e_1$, but $g$ is not the direct sum of $h$ and a $p$-unipotent $p$-subalgebra.

(d) If $g$ is $p$-unipotent, show that in the restricted enveloping algebra of $g$ the two-sided ideal generated by $g$ is nilpotent. (Argue by induction on the dimension of $g$.)