Corollary

Let $y \in X$. The free Lie algebra $L(X)$ is the direct sum of the free submodule $K.y$ and the Lie subalgebra admitting as basic family the family of $((\mathrm{ad}\ y)^n, z)$ where $n \geqslant 0$ and $z \in X - \{y\}$.

It suffices to put $S = \{y\}$ in Proposition 10.