Exercise 14, § 2

Suppose that $K$ is a field. Let $X$ be a set and $x, y$ two elements of $L(X)$ which are 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 $x$ (resp. $y$) of highest degree. Adding if necessary a multiple of $y$ to $x$, we can assume that $x_p$ and $y_q$ are linearly independent. The subalgebra of $L(X)$ generated by $x_p$ and $y_q$ is graded and hence free, cf. Exercise 13. Deduce that $(x_p, y_q)$ is a free family and pass from this to $(x, y)$.)