How to obtain division algebras from a generalized Cayley-Dickson doubling process

New families of eight-dimensional real division algebras with large derivation algebra are presented: We generalize the classical Cayley-Dickson doubling process, starting with a quaternion algebra over a field F and allowing the element used in the doubling to be an invertible element in the algebra. The resulting unital algebras are not third power-associative, hence not quadratic. Starting with a quaternion division algebra D, we obtain division algebras A for all elements chosen in D outside of F. This is independent of where the element is placed inside the product. Thus three pairwise non-isomorphic families of eight-dimensional division algebras are obtained. Their Albert isotopes yield more division algebras with large derivation algebra.