Exercise 12, § 2

Let $X = \{x, y\}$ be a set with two elements. Prove that the derived algebra of $L(X)$ is a free Lie algebra, admitting as basic family the elements $((\operatorname{ad} y)^q \circ (\operatorname{ad} x)^p)(y)$, for $p \geq 1,\ q \geq 0$. (Use Ex. 11.)