Remark

Suppose that $X$ consists of two distinct elements $x$ and $y$ and let $L^{(\cdot,1)}$ be the submodule of $L(X)$ the sum of the $L^\alpha(X)$ where $\alpha \in \mathbf{N}^X$ and $\alpha(y)=1$. It follows immediately from Theorem 1 and Proposition 12 of no. 10 that the elements of $(\mathrm{ad}\,x)^n.y$ where $n$ is an integer $\geqslant 0$ form a basis of the submodule $L^{(\cdot,1)}$. It follows that the restriction to $L^{(\cdot,1)}$ of the mapping ad $x$ is injective.