Corollary 2

Let $a$ be a Lie algebra. Every extension of $L(X)$ by $a$ is inessential.

Let $a \xrightarrow{\lambda} g \xrightarrow{\mu} L(X)$ be such an extension (Chapter I, § 1, no. 7). As $\mu$ is surjective, there exists a mapping f of X into $g$ such that $\phi = \mu \circ f$. Let F be the homomorphism of $L(X)$ into $g$ such that $f = F \circ \phi$ (Proposition 1). Then $(\mu \circ F) \circ \phi = \mu \circ f = \phi$ and Proposition 1 shows that $\mu \circ F$ is the identity automorphism of $L(X)$. The given extension is therefore inessential (Chapter I, § 1, no. 7, Proposition 6 and Definition 6).

As the ring K is non-zero, Corollary 1 to Proposition 1 shows that $\phi$ is injective. Hence the set X can be identified by means of $\phi$ with its image in $L(X)$; with this convention, X generates $L(X)$ and every mapping of X into a Lie algebra $g$ can be extended to a Lie algebra homomorphism of $L(X)$ into $g$.