Corollary 2

Tag 0AC9, permanent. Cite it as [Bourbaki, Tag 0AC9].

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

If the mappings $f$ and $g$ of $M$ into $G$ have the same left differential, the tangent mapping to $fg^{-1}$ is everywhere zero. If further $K$ is of characteristic 0, then $fg^{-1}$ is locally constant.

By Proposition 59,

(fg^{-1})^{-1} \cdot d(fg^{-1}) = (\mathrm{Ad} \circ g \circ p) \circ (f^{-1}.df) - (\mathrm{Ad} \circ g \circ p) \circ (g^{-1}.dg).

If $f^{-1}.df = g^{-1}.dg$ , then $(fg^{-1})^{-1} \cdot d(fg^{-1}) = 0$ , that is $T_x(fg^{-1}) = 0$ for all $x \in M$ . This proves the first assertion. The second follows from it by Differentiable and Analytic Manifolds, R, 5.5.3.

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

Cited by

Proposition 15, § 6

Cites

Proposition 59