Exercise 27, § 6

(a) Show that, in a finite-dimensional Lie $p$-algebra, the radical is a $p$-ideal. (Use Exercise 22 (c) of § 1.)

(b) For a Lie $p$-algebra $g$ to have zero radical, it is necessary and sufficient that $g$ contain no commutative $p$-ideal $\neq \{0\}$. (Use Exercise 22 (c) of § 1.)