Remark

When X is empty, $M(X)$ is empty and hence $L(X) = \{0\}$. If X consists of a single element $x$, the submodule $K.x$ of $L(X)$ is a Lie subalgebra of

L(X); as X generates L(X), Corollary 1 to Proposition 1 shows that L(X) is a free module with basis {x}.