On Asano's theorem

Let D be a division algebra over an infinite field K such that every element of D is a sum of finitely many algebraic elements. As a generalization of Asano's theorem, it is proved that every noncentral subspace of D invariant under all inner automorphisms induced by algebraic elements contains [D, D], the additive subgroup of D generated by all additive commutators of D. Flom the viewpoint we study the existence of normal bases of certain subspaces of division algebras. It is proved among other things that D is generated by multiplicative commutators as a vector space over the center of D.