Remark 2

Tag 0DN9, permanent. Cite it as [Bourbaki, Tag 0DN9].

Algebra, chapter IX, § 8. Types de formes quadratiques

Since $A$ is of characteristic $\neq 2$ , every element $T$ of $\mathfrak{B}$ can be written in the form $\sum_{i=1}^n a_i . T_1$ where $a_i \in A^*$ (No. 2, prop. 4). One has

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(\sum_{i=1}^n a_i . T_1)(\sum_{j=1}^q b_j . T_1) = \sum_{i,j} a_i b_j . T_1 \qquad (a_i, b_j \text{ dans } A^*).

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

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