When I took abstract algebra as an undergrad, we did a brief bit on the quaternions. Bursting with curiosity I asked the professor if 8 and 16 dimensional structures existed. "Of course! But just as you lose commutivity with Q, when you go to the octonions, you lose associativity, and the sedonions lack "alternativity" (had to look that up -- I didn't remember) and they're basically algebraic novelties with out any application."
Right, but while the Cayley-Dickson construction mostly provides novelties (though I remember reading something about octonions and string theory[1]), Clifford algebras are derived differently; they are isomorphic to complex numbers and quaternions for two and three base vectors respectively, but they produce something else after quaternion. This "something different" can be used to represent, you guessed it, reflections and rotations in a 4D space. Because they are not obtained from the Cayley-Dickson construction they are not division algebras, however.
