Why are properties lost in the Cayley–Dickson construction?

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.

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Not directly an answer, but the paper math.usask.ca/~bremner/research/publications/BHiaocdp.pdf may be of interest. All the algebras constructed from the Cayley-Dickson process satisfy $(xy)x = x(yx)$ and $x^2 (yx) = (x^2 y)x$. In particular, they are power associative, i.e., (nonnegative integer) powers of elements are well-defined. –  Ted Feb 29 '12 at 17:12
See also what John Baez has to say: math.ucr.edu/home/baez/octonions/node5.html –  Yuval Filmus Jun 25 '12 at 3:48

There are eight equivalent definitions of the Cayley-Dickson product of ordered pairs, but one that is commonly used is $(a,b)(c,d)=(ac-db^*,a^*d+cb)$ where the conjugate (for all eight variations) is defined as $(a,b)^*=(a^*,-b)$.

The unit basis vectors $e_0,e_1,e_2,\cdots$ for all finite dimensional Cayley-Dickson vectors may be defined in various ways, but to preserve the inherent symmetry of the multiplication table, the most natural way to define the sequence is $e_{2n}=(e_n,0)$ and $e_{2n+1}=(0,e_n)$ for $n\ge0$. Note that this produces a different numbering from the usual numbering of the Octonion basis vectors as used by Octonion specialists and, that as a result, their multiplication table does not reveal the inherent symmetry of the process.

In addition to revealing the symmetry of the Cayley-Dickson process, this numbering of the basis vectors has the added advantage that for all non-negative integers $i,j$ it is true that $e_ie_j=\pm e_k$ where $k$ is the bit-wise "exclusive or" of the binary representations of $i$ and $j$.

Using this numbering of the basis vectors, the multiplication table of any finite dimensional Cayley-Dickson space can be recovered from the following properties:

1. $e_1e_{2n}=e_{2n+1}$ for all $n\ge0$.
2. If $0\ne i\ne j\ne 0$ then $e_ie_j=e_k$ implies all the following:
1. $i\ne k$
2. $j\ne k$
3. $k\ne 0$
4. $e_je_i=-e_k$
5. $e_je_k=e_i$ (these last two are the quaternion properties)
6. $e_{2i}e_{2j}=e_{2k}$
7. $e_{2i}e_{2j+1}=e_{2k+1}$
8. $e_{2i+1}e_{2j}=e_{2k+1}$
9. $e_{2j+1}e_{2i+1}=e_{2k}$ (note the reversal of $i$ and $j$)

For four of the other seven alternate ways to define the Cayley-Dickson product, property 1 differs, and for all seven, some of the rules 2.7, 2.8 and 2.9 differ.

Now whether the symmetry of the multiplication tables will explain the loss of properties, I do not know, but it might be an interesting subject for someone to investigate.

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Usefulness of an algebra isn’t determined by symmetries of multiplication table. Moreover, there are no obstacles to produce associative (and even commutative) algebras of arbitrary dimension and we do not need Cayley–Dickson construction for it. But a crudely constructed associative algebra has great chances to run onto zero divisors. It might, in turn, indicate that we haven’t anything new and the algebra is isomorphic to something dull. One option is the ring of n × n matrices over the ground field. The ring of real 2 × 2 matrices has, for example, an alternative name: split-quaternions. How many readers heard of it before? Almost any guy knows such terms as complex numbers and quaternions, that are not “simply” real matrix rings, but only hypercomplex fans want to invent special terms for square matrices, although their “multiplication table” is very symmetric.

Another possibility is to “find” a direct sum of certain lesser-dimesional algebras (see, for example, this drama in English Wikipedia). Only cranks are eager to discover reducible objects and assign names to them. Mathematicians, of course, use them sometimes, but bearing in mind that they are reducible.

Why is Cayley–Dickson construction famous, indeed? It is an heuristic that is proven to discover something new at its first 3 steps, and happily avoid zero divisors. First time it made the only possible finite-dimensional extension field of ℝ. Second time it made the only remaining finite-dimensional skew field. Third time it discovered the only 8-dimensional division algebra. There should be no question why do algebras degrade, because the supply of good algebras is very limited. There should be a question how the modern formulation of “Cayley–Dickson construction” makes necessary precautions to make this inevitable degradation graceful at each step. Key points are:

• Multiplicative identity is always preserved;
• Commutativity is preserved, given * = id in the original ring;
• Associativity is preserved, given a commutative original ring;
• From a *-ring satisfying certain conditions it makes an alternative algebra.

Could we cheat the construction? Well… give it, for example, ℂ with identical “*” instead of complex conjugation. You’ll obtain a commutative algebra over ℂ, but without division, because there are no finite-dimensional complex division algebras (but ℂ); there are only real ones. Note that there is no general algebraic substantiation for absence of zero divisors. We were lucky to advance so far, when started from ℝ, without running on it.

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