Remark

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

Algebra, chapter IX, § 4. Sous-espaces totalement isotropes. Théorème de Witt

Under the conditions of Corollary 1, E is the direct sum of $F + F'$ and the orthogonal of $F + F'$ . We therefore have a Witt decomposition of E. By prop. 2, a), there exist bases of F and F’ such that, in matrix (1) of $\Phi$ , the block $U$ is the unit matrix $1_r$ .

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

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