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 find representation as matrices over F?
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You could take the regular representation (left multiplication on $A$). So if $x^2 = a, y^2 = b$ then taking a basis $\{1,x,y,xy\}$ of $A$, $x$ would be represented by the matrix $$\left( \begin{array}{cccc} 0 & a & 0 & 0 \cr 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & a \cr 0 & 0 & 1 & 0 \end{array} \right), $$ etc. 

