Corollary 2
Tag 0DL1, permanent. Cite it as [Bourbaki, Tag 0DL1].
Algebra, chapter IX, § 4. Sous-espaces totalement isotropes. Théorème de Witt
The hypotheses being those of Theorem 1, the group of metric automorphisms of E permutes transitively the totally isotropic (resp. totally singular) subspaces of given dimension of E. Moreover, if F is a totally isotropic (resp. totally singular) subspace of E, every bijective linear mapping of F onto F is induced by a metric automorphism of E.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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