I am looking for a proof not using Wedderburn-Artin nor Skolem-Noether of:
Every central simple (division) algebra of dimension 4 over $F$ (a field of characteristic $\neq 2$) is quaternion.
All the proofs that I found use either Wedderburn-Artin and/or Skolem-Noether and I can't figure out an alternative proof.
I found a proof here, but they also use Skolem-Noether.
Thank you in advance.
A proof is also asked in Exercise $15$ on page $136$ in the book Noncommutative Algebra by Benson Farb and R. Keith Dennis here. The hint is as follows: See the proof of the Frobenius Theorem. So you could follow this proof in the book. On the other hand I do not see why we should avoid Wedderburn, or Skolem-Noether. These are very basic results in this area which one should study.