Quadratic division algebras revisited (Remarks on an article by J. M. Osborn)

In his remarkable article "Quadratic division algebras" (Trans. Amer. Math. Soc. 105 (1962), 202-221), J. M. Osborn claims to solve `the problem of de termining all quadratic division algebras of order 4 over an arbitrary fiel d F of characteristic not two ... modulo the theory of quadratic forms over F' (cf. p. 206). While we shall explain in which respect he has not achiev ed this goal, we shall on the other hand complete Osborn's basic results (b y a reasoning which is finer than his) to derive in the real ground field c ase a classification of all 4-dimensional quadratic division algebras and t he construction of a 49-parameter family of pairwise nonisomorphic 8-dimens ional quadratic division algebras. To make these points clear, we begin by reformulating Osborn's fundamental observations on quadratic algebras in categorical terms.