Proposition 18
Tag 0AFY, permanent. Cite it as [Bourbaki, Tag 0AFY].
Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups
Let E be a complete normable space over C, v ∈ L(E) and g = exp v. Suppose that Sp(v) contains none of the points 2iπn with n ∈ Z − {0}. Then, for all x ∈ E, the conditions vx = 0 and gx = x are equivalent.
This follows from Lemma 2 of no. 4, applied to the function z ↦ e^z − 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.