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Representation quaternions as matrices

char F≠2, a,b invertable from F, A(a,b) - generalised quaternions. Using Artin–Wedderburn theorem there is a representation of them over F. I found representation as Q8 but it's not over F. So, how to ...
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Is there a good version of Artin-Wedderburn for semisimple algebra objects?

Artin-Wederburn says that if you have a semisimple algebra then it is a product of matrix algebras over division rings. Suppose that C is a fusion category over the complex numbers (if you want to ...