Osborn has shown that any
quadratic algebra over a field of characteristic not two can be decomposed into a copy
of the field and a skewcommutative algebra with a bilinear form. For any
nonassociative algebra G over a field of characteristic not two, Albert and Oehmke
have defined an algebra over the same vector space, which is bonded to G by a linear
transformation T. In this paper this process is specialized to the class 𝒜 of
finite dimensional quadratic algebras A over fields of characteristic not two,
which define a symmetric, nondegenerate bilinear form, to obtain quadratic
algebras B(A,T) bonded to A. In the main results T will be defined as a linear
transformation on the skewcommutative algebra V defined by Osborn’s
decomposition of A. An algebra in 𝒜 is called a division algebra if A≠0 and the
equations ax = b and ya = b, where a≠0 and b are elements in A, have unique
solutions for x and y in A. Consequently, a finite dimensional algebra A≠0 is a
division algebra if and only if A has no divisors of zero. A basis for V is said
to be orthogonal, if it is orthogonal with respect to the above mentioned
bilinear form. An algebra in 𝒜 is weakly flexible if the ith component of the
skewcommutative product of the ith and jth members of each orthogonal basis of
V is 0. If 𝒟(𝒜) denotes the class of division algebras in 𝒜 and I denotes the
identity transformation on V , then the main results are: (1) A ∈𝒟(𝒜), T
nonsingular and B(A,T) flexible imply B(A,T) ∈𝒟(𝒜), (2) if A ∈𝒟(𝒜) and A is
weakly flexible, then B(A,T) is weakly flexible if and only if T = δI for δ a
scalar, and (3) if A is a CayleyDickson algebra in 𝒟(𝒜), then B(A,T) is a
CayleyDickson algebra in 𝒟(𝒜) if and only if T = ±I. Finally, a class of nonflexible
quadratic division algebras bonded to CayleyDickson division algebras will be
exhibited.
