Lemma 3

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

Algebra, chapter IX, § 9. Algèbres de Clifford

Let F and G be two bilinear forms on E. We have $\lambda_F \circ \lambda_G = \lambda_{F+G}$ . For every bilinear form F on E, $\lambda_F$ is a bijection of $T(E)$ onto itself.

Indeed, $\lambda_F \circ \lambda_G$ has the characteristic properties (7) and (8) of $\lambda_{F+G}$ : we have $(\lambda_F \circ \lambda_G)(1) = 1$ , and

\begin{align*} \lambda_F \circ \lambda_G \circ e_x &= \lambda_F \circ (e_x + i_x^G) \circ \lambda_G = (e_x + i_x^G + i_x^F) \circ \lambda_F \circ \lambda_G \\ &= (e_x + i_x^{F+G}) \circ \lambda_F \circ \lambda_G. \end{align*}

On the other hand, if $F = 0$ , we have $i_x^F = 0$ for every $x \in E$ , and consequently $\lambda_F = I$ , which entails that, for every F, we have $\lambda_F \circ \lambda_{-F} = \lambda_{-F} \circ \lambda_F = I$ .

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

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Cites

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