Remark 1

Suppose that the characteristic of K is different from 2, and set $i'= 2i-\beta$. Then $(1, i')$ is a basis of E over K, and we have ${i'}^2= 4\alpha +\beta^2$. It follows that E is isomorphic to the quadratic algebra of type $(4\alpha +\beta^2,0)$ and F to the quaternion algebra of type $(4\alpha +\beta^2,0, \gamma )$.