symmetric monoidal (∞,1)-category of spectra
An operation satisfying one of these conditions is left-alternative, flexible, or right-alternative, respectively, and it is alternative if it is both left-alternative and right-alternative. A magma is so if its binary operation is so, and a nonassociative algebra (or nonassociative ring?) is so if its multiplication operation is so.
A commutative? operation/magma/algebra must be flexible, and it is left-alternative iff it is right-alternative (and so simply alternative). An associative? operation/magma/algebra is both alternative and flexible.
Among algebras (but not magmas), any algebra with two of these three properties must have the third. In particular, an alternative algebra must be flexible. This follows from the characterization in terms of the associator below.
is alternating, in that whenever two of the three arguments are equal, the result is zero. Over a field whose characteristic is different from , then, or more generally over any commutative ring in which is invertible or even cancellable, alternativity is equivalent to the condition that the associator is skew-symmetric?, in that for any permutation of three elements then for the signature of the permutation.
In one direction, alternativity together with the multi-linearity of the product says that the associator is skew symmetric in adjacent arguments
and hence in all arguments.
In the other direction, skew-symmetry of the associator implies alternativity by
using the assumption that is cancellable in . Similarly for .
This is due to (Zorn 30).
Every Cayley–Dickson algebra over a commutative ring is flexible. The first three (corresponding, if we start with the real numbers, to the real numbers, complex numbers, and quaternions) are associative and hence alternative. The next one (corresponding to the octonions) is still alternative despite not being associative (unless has characteristic ). After that (corresponding to the sedenions and above), they are not even alternative (unless has characteristic ).
R. D. Schafer, chapter III of Introduction to Non-Associative Algebras Dover, New York, 1995. (web)
Max Zorn, Theorie der alternativen Ringe, Abhandlungen des Mathematischen Seminars der Universität Hamburg 8 (1930), 123-147