Let $K$ be a field and $D$ be a central division algebra over $K$ of degree $n$. Suppose that $L\subset D$ is a maximal subfield, so that $[L:K]=n$. Then we know that $L$ is a splitting field, so there exists an isomorphism $f:D\otimes_KL\to M_n(L)$ of $L$-algebras.
My question is: does it always exists an $f$ as above such that $f(L\otimes _KL)$ is the set of diagonal matrices in $M_n(L)$?
(asked in https://math.stackexchange.com/q/1619116/217671 but I think it might be more appropriate here).