Remark 5

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

Algebra, chapter IX, § 1. Formes sesquilinéaires

Let $A$ be a ring endowed with an antiautomorphism $J$ , $E$ a left $A$ -module, $G$ an $(A, A)$ -bimodule and $\Phi$ a mapping from $E \times E$ into $G$ , right sesquilinear for $J$ . Prove the identity (where $Q(x) = \Phi(x, x)$ ) :

2\Phi(x, y)\ (\mu^J\lambda^J - \lambda^J\mu^J) = Q(x - \mu\lambda y) - Q(x + \mu\lambda y) + \mu Q(x + \lambda y) - \mu Q(x - \lambda y) + Q(x + \mu y)\lambda^J - Q(x - \mu y)\lambda^J + \mu Q(x - y)\lambda^J - \mu Q(x + y)\lambda^J.

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

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