Corollary

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

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

The canonical mapping $\zeta$ of $L(A^*)$ into $A$ is an isomorphism of the Lie algebra, $L(A^*)$ onto the Lie algebra $A$ . In other words,

$\zeta([a, b]) = \zeta(a)\zeta(b) - \zeta(b)\zeta(a)$

for all $a, b$ in $L(A^*)$ . If $K$ is of characteristic $p > 0$ , then $\zeta(a^p) = \zeta(a)^p$ for all $a \in L(A^*)$ .

Henceforth $L(A^*)$ and $A$ are identified by means of the isomorphism $\zeta$ .

The canonical morphism of $\mathscr{T}^{(\infty)}(A^*)$ into $A$ has been obtained as a special case of the morphism of Proposition 33. But it is possible to argue in the opposite direction:

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

Cited by

§ 3. Passage from a Lie group to its Lie algebra
Example, § 4

Cites

Proposition 33