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I'm studying Clifford algebras on this moment, and I frequently find the statement $$\left(\mathbb{R}_m^{(2)},[\cdot,\cdot]\right) \cong \mathfrak{so}_{\mathbb{R}}(m)$$ stating that the bivectors of a real Clifford algebra are isomorphic to the real special orthogonal Lie algebra. Unfortunately I'm not able to find the original theorem proving this.

Is there anyone that can help me with the finding of this theorem? Or is there an easy way to see this myself?

Thanks in advance

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up vote 1 down vote accepted

There is a proof in Chapter 5 of Gracia-Bondia, Varilly, and Figueroa's book Elements of Noncommutative Geometry. They prove in Lemma 5.7, page 182 the result that bivectors in the Clifford algebra $Cl(V)$ are closed under taking commutators, and that the adjoint action of bivectors on vectors in $V$ induces an isomorphism of the Lie algebra of bivectors with the real orthogonal Lie algebra $\mathfrak{so}(V)$. This is what you want, I think.

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I hope you don't mind: I went ahead and just changed it to mathfrak for broader compatibility. – rschwieb Mar 5 '14 at 19:35
No, I don't mind. Thanks very much ! – Dietrich Burde Mar 5 '14 at 19:41

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