Corollary 1
Tag 0ABD, permanent. Cite it as [Bourbaki, Tag 0ABD].
Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra
Let u∈L(G) and u′∈T(∞)(G). Then u⊤u′=u∗u′−u′∗u.
The image of u under the coproduct is u⊗εe+εe⊗u, whence
u⊤u′=u∗u′∗εe+εe∗u′∗u∨=u∗u′−u′∗u.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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