I asked this quetion five days ago at https://math.stackexchange.com/questions/406669/are-the-elements-of-a-division-algebra-which-commute-with-all-commutators-in-the Some good people have given good comments there.
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$\begingroup$ This is a lemma in Ancohea's paper jstor.org/discover/10.2307/…. I can not understand the proof. $\endgroup$– yanyuCommented Jun 3, 2013 at 4:43
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$\begingroup$ 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 $\endgroup$– user30180Commented Jun 3, 2013 at 6:22
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1 Answer
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Yes because the division ring generated by commutators is invariant under all inner automorphisms and the result follows from Cartan-Brauer-Hua.
