Exercise 14, § 2

Suppose that $K$ is a field. Let $X$ be a set, and let $x, y$ be two elements of $L(X)$ linearly independent over $K$. Show that the family $(x, y)$ is free in the Lie algebra $L(X)$. (Let $x_p$ (resp. $y_q$) be the non-zero homogeneous component of highest degree of $x$ (resp. of $y$). By adding to $x$ a multiple of $y$, we may suppose that $x_p$ and $y_q$ are linearly independent. The subalgebra of $L(X)$ generated by $x_p$ and $y_q$ is graded, hence free, cf. Exercise 13. Deduce that $(x_p, y_q)$ is a free family and pass from there to $(x, y)$.)