I am looking for papers or books which discuss this problem. Thank you for reading:

Let K and L be two subfields of a non-commutative division algebra D with the center Z. Suppose that K and L contain Z and that D is finite dimensional over Z. Let V be the tensor product of K and L over Z, viewed as a vector space over Z. Consider the linear map g from V to D defined by g(x \otimes y) = xy, for every x in K and y in L. My question is this:

Under what condition(s) on K and L the map g is injective?

What is clear is that the intersection of K and L has to be Z. But that is not sufficient in general. Would this necessary condition be also sufficient if K and L were maximal subfields?