Why are there no Dual-octonions?

Question

In the case of quaternions, we can define the traditional quaternions setting the imaginary components equal to root negative one, the hyperbolic quaternions by using root positive one, and the dual quaternions by using root zero. Actually, this construction works in the planar case as well, producing the unit circle, unit hyperbola, and unit semi circle respectively. I would naively expect this to be true for the octonions as well, but I read that only the traditional and split octonions exist. Why?

Answer 1

You're asking a question in which people see common mystical roots in the Tetragrammaton of Paleo-Hebrew, the Tetractys of the Greeks and the quaternion worship of the Victorians. In another sense, there can certainly be dual octonions and complex octonions. I just thought them up myself (independently, of course.) They can certainly exist, but they lack some of the properties that make algebra useful, and they're just not useful for anything.

Historically anyway, the usefulness of an algebra has been measured in its capacity for expressing and manipulating geometry and mechanics, and the useful ones have fallen into a set of the only four normed division Clifford algebras. They have the dimensions 2^0=1, 2^1=2, 2^2=4 and 2^3=8, corresponding to the reals, complex, quaternions and octonions.

The reals express and can be used to transform magnitudes, the complex to express and transform quantities in 2D space, the quaternions to express direction and spherical rotations in 3D space and the octonions to add spatial position and spatial transformations. The octonion form underlies Hestenes' geometric algebra, maybe Grassmann's too and the dual quaternions.

So what more could you want anyway?