If $L$ splits $D=I\otimes N$ does $L$ split $I$ and $N$?

Question

Let $D$ be a central simple division algebra over the field $F$. Let $D\sim I\otimes N$ where $I$ and $N$ are division algebras in Br$(F)$ and $\sim$ is equivalence in Br$(F)$. I am interested in this question.

If $L$ splits $D$ does $L$ split $I$ and $N$?

I can't seem to find this in any of the literature. Every example I can think of holds true. Does anyone know of a proof, have a reference, or a counter example? Any of them would be great.

Answer 1

Certainly not:
Take any non-trivial division algebra $I$ over $F$ and let $N=I^{opp}$.
Then $F\sim I\otimes_F N$ is trivial in $Br(F)$ hence is split by $F$.
However neither $I$ nor $N$ is split by $F$.