I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ dimensions. I think I understand the first $n<4$ instances:

- As we move from $\mathbb{R}$ to $\mathbb{C}$ we lose ordering
- From $\mathbb{C}$ to $\mathbb{H}$ we lose the commutative property
- From $\mathbb{H}$ to $\mathbb{O}$ we lose the associative property (in the form of $(xy)z \neq x(yz)$, but apparently it's still alternative and $(xx)y = x(xy)$. Is that right?)
- The move from $\mathbb{O}$ to $\mathbb{S}$ is where I start to get fuzzy. From what I've read, the alternative property is broken now, such that even $(xx)y \neq x(xy)$ but that also zero divisors come into play, thus making sedenion algebra non-division.

My first major question is: Does the loss of the alternative property cause the emergence of zero divisors (or vice versa) or are these unrelated breakages?

My bigger question is: What specific algebraic properties break as we move into 32 dimensions, then into 64, 128, 256? I've "read" the de Marrais/Smith paper where they coin the terms pathions, chingons, routons and voudons. At my low level, any initial "reading" of such a paper is mostly just intent skimming, but I'm fairly certain they don't address my question and are focused on the nature and patterns of zero divisors in these higher dimensions. If the breakages are too complicated to simply explicate in an answer here, I'm happy to do the work and read journal articles that might help me understand, but I'd appreciate a pointer to specific papers that, given enough study, will actually address the point of my specific interest--something I can't necessarily tell with an initial glance, and might need a proper mathematician to point me in the right direction.

Thank you!

UPDATE: If the consensus is that this is a repeat, then ok, but I don't see how the answers to the other question about *why* algebraic properties break answers my questions about *what* algebraic properties break. Actually, the response marked as an answer in that other question doesn't actually answer that question either. It provides a helpful description of how to construct a multiplication table for higher dimension Cayley-Dickson structures, but explicitly doesn't answer the question as to why the properties break.

The Baez article many people suggest in responses to all hyper-complex number questions like mine is truly excellent, but is mostly restricted to octonions, and, in the few mentions it makes of higher dimension Cayley-Dickson algebras, does not refer to what properties are broken.

Perhaps the question isn't answerable, but in any case it hasn't been answered in this forum.

UPDATE 2: I should add that the sub question in this post about whether the loss of the alternative property specifically leads to the presence of zero divisors in sedenion algebra is definitely unique to my question. However, perhaps I should pose that as a separate question? Sorry, I'm not sure about that aspect of forum etiquette here.