Triviality of the functor Coker(K₁(F) → K₁(D)) for division algebras

Let D be a division algebra with center F. Consider the group CK ₁(D) = D*/F*D′ where D* is the group of invertible elements of D and D′ is its commutator subgroup. In this note we shall show that, assuming a division algebra D is a product of cyclic algebras, the group CK¹(D) is trivial if and only if D is an ordinary quaternion algebra over a real Pythagorean field F. We also characterize the cyclic central simple algebras with trivial CK₁ and show that CK ₁ is not trivial for division algebras of index 4. Using valuation theory, the group CK₁(D) is computed for some valued division algebras.