How to Obtain Non-Flexible Quadratic Algebras as Mutation Algebras of Hurwitz Algebras

We present an elementary way to obtain new families of division algebras of degree n out of division algebras with a multiplicative norm. Families of four- and eight-dimensional non-flexible quadratic division algebras over a field F of characteristic not 2 are easily constructed as a-mutation algebras of Hurwitz division algebras A over F, choosing ${a \in A \setminus F}$ suitable. In particular, we thus obtain real eight-dimensional division algebras with large derivation algebras.