We investigate the class of finite-dimensional real flexible division algebras. We classify the commutative division algebras, completing an approach by Althoen and Kugler. We solve the isomorphism problem for scalar isotopes of quadratic division algebras, and classify the generalised pseudo-octonion algebras. In view of earlier results by Benkart, Britten and Osborn and Cuenca Mira et al., this reduces the problem of classifying the real flexible division algebras to the normal...

The concept of doubling, which was introduced around 1840 by Graves and Hamilton, associates with any quadratic algebra 𝓐 over a field k of characteristic not 2 its double 𝓥(𝓐 ) = 𝓐 × 𝓐 with multiplication (w,x)(y,z) = (wy - z̅x,xy̅ + zw). This yields an endofunctor on the category of all quadratic k-algebras which is faithful but not full. We study in which respect the division property of a quadratic k-algebra is preserved under doubling and, provided this is the case, whether the...

The algebras ℂ (complex numbers), ℍ (quaternions), and 𝕆 (octonions) are real division algebras obtained from the real numbers ℝ by a doubling procedure called the Cayley-Dickson Process. By doubling ℝ (dim 1), we obtain ℂ (dim 2), then ℂ produces ℍ (dim 4), and ℍ yields 𝕆 (dim 8). The next doubling process applied to 𝕆 then yields an algebra 𝕊 (dim 16) called the sedenions. This study deals with the subalgebra structure of the sedenion algebra 𝕊 and its zero divisors. In particular, it shows...