Corollary 2

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

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

Let $t \in \mathscr{T}^{(\infty)}(G)$ and $g \in G$ . Then $\varepsilon_g \top t = \varepsilon_g * t * \varepsilon_{g^{-1}}$ . If $t \in \mathbf{L}(G)$ , then $\varepsilon_g \top t = gtg^{-1}$ (where the latter product is evaluated in the group $\mathbf{T}(G)$ ).

The image of $\varepsilon_g$ under the coproduct is $\varepsilon_g \otimes \varepsilon_g$ .

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

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