Proposition 7

Tag 09U2, permanent. Cite it as [Bourbaki, Tag 09U2].

Lie Groups and Lie Algebras, chapter I, § 1. Definition of Lie algebras

Let $a$ and $b$ be two Lie algebras over $K$ ,

a \xrightarrow{\lambda} g \xrightarrow{\mu} b

an inessential extension of $b$ by $a$ , $\nu$ an isomorphism of $b$ onto a subalgebra of $g$ such that $\mu \circ \nu$ is the identity automorphism of $b$ and $\phi$ the corresponding homomorphism of $b$ into the Lie algebra of derivations of $a$ . Let

a \xrightarrow{\lambda_0} g_0 \xrightarrow{\mu_0} b

be the inessential extension of $b$ by $a$ canonically defined by $\phi$ . Then the mapping $(a, b) \mapsto \lambda(a) + \nu(b)$ is an isomorphism $f$ of $g_0$ onto $g$ and the following diagram

\begin{array}{ccccc} & & \\ a & \xrightarrow{\lambda_0} & g_0 & \xrightarrow{\mu_0} & b \\ & \searrow f & \downarrow & \nearrow \mu_0 \\ & & g & & \\ & \swarrow i & & \searrow \mu \\ & & & & \end{array}

is commutative, so that the two extensions are equivalent.

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

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