Bourbaki

Proposition 21

Tag 0AA1, permanent. Cite it as [Bourbaki, Tag 0AA1].

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

Let ψ\psi be an automorphism of X commuting with the operations of G. Then DtD_t is invariant under ψ\psi.

For all x∈Xx \in X,

(ψ(Dt))(ψ(x))=ψ∗(Dt(x))=ψ∗(t∗εx)=t∗ψ∗(εx)=t∗εψ(x)=Dt(ψ(x)). \begin{align*} (\psi(D_t))(\psi(x)) &= \psi_*(D_t(x)) = \psi_*(t * \varepsilon_x) \\ &= t * \psi_*(\varepsilon_x) \tag{Proposition 15} \\ &= t * \varepsilon_{\psi(x)} = D_t(\psi(x)). \end{align*}

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

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Transcribed from the printed volume. Not translated.