In the book Skew Fields, by P.K.Draxl, at page 60, states there as lemma 2:
Let $A,B,C$ be finite-dimensional $K$-algebras such that $|C:K|\leq|A:K||B:K|$ and let $f: A\rightarrow C$, and $g: B\rightarrow C$ be $K$-algebra homomorphisms. Then $A \otimes B \cong C$ provided $A,B$ are central simple $K$-algebras.
There it is alluded to theorem 2 in section 5, where one states that if, notations as above, f and g commute, then there will be an R-algebra homomorphism from $A \otimes B$ to C. But, thus far as I observe, there appears no such condition in the statements of the lemma, so I could not draw conclusions from the theorem, could I?
Any help would be greatly appreciated.
Thanks for any attention.