Remark 1
Tag 0AXE, permanent. Cite it as [Bourbaki, Tag 0AXE].
Lie Groups and Lie Algebras, chapter VI, § 2. Affine Weyl group
The assumption that T is of rank l is satisfied in particular when G is irreducible and infinite. Indeed, the vector space generated by vectors corresponding to the translations in T is invariant under the canonical image of G in the linear group of F. It is different from if G is infinite and is thus equal to the whole of F if G is infinite and irreducible.
A finite group generated by reflections is not always the Weyl group of a root system. More precisely:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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