Corollary

Tag 0A37, permanent. Cite it as [Bourbaki, Tag 0A37].

Lie Groups and Lie Algebras, chapter II, § 2. Free Lie algebras

Every mapping of $X$ into $L(X)$ can be extended uniquely to a derivation of $L(X)$ .

When $M$ is equal to $L(X)$ with the adjoint representation, relation (20) means that $D$ is a derivation.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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