**Motivating question**: What lies beyond the Sedenions?

I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process:

$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$

"Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions" $\subset$ $\ldots$

and that at each step you're given a multiplication table that tell how the elements interact. As you move up the ladder, certain "nice" properties are lost: ordering, commutativity, associativity, multiplicative normedness, etc... Given the multiplication table, you can show that these properties don't hold.

Eric Naslund noted that "the first 4 are very special as they are the unique 4 normed division algebras over ℝ", no surprise then that these $2^n$-ions have found quite a bit of use. I'm interested in the sequence itself however, irrespective of how useful a $2^{256}$-ion might be (*ducenti-quinquaginta-sex-ion*?).

I feel like something deeper is going on here though that I don't understand. Why are these particular properties lost at each step? Is it possible to quantify the process such that, at the $2^n$-ion you can say something about the symmetry of the multiplication table*?

* I'm making an ansatz that there is a connection between the symmetry of the multiplication table and these "nice" properties.