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This is a lemma in Ancohea's paper…. I can not understand the proof. – yanyu Jun 3 '13 at 4:43
Hmm, the proof of the lemma in that paper seems to suggest Ancohea assumes the division algebra is finite-dimensional over the center, but this is never clearly stated earlier in the paper. How about the following 1-sentence proof: the commutators over $k$ span the space of commutators over $\overline{k}$, so it suffices to treat the case of a matrix algebra, which you can treat by bare hands. QED – user30180 Jun 3 '13 at 6:22

Yes because the division ring generated by commutators is invariant under all inner automorphisms and the result follows from Cartan-Brauer-Hua.

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