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The Cayley–Dickson construction is a method of generalizing the algebraic structures of the Reals to the Complex to the Quaternions, etc.

For those interested, discussion on the algebraic properties Cayley–Dickson constructions are not new here. For instance, see these older posts:

The first is about new properties gained in each iteration, and the second is about why properties are lost in each iteration. I hope it becomes evident why my question differs (albeit subtly) from previous questions.

Cracking this case open once more, what I find most intriguing about this mystery can be briefly explained by this chart, explaining the algebraic properties of each level of the construction:

Cayley-Dickson algebraic properties

Not surprisingly, we should expect these types of properties to drop off as the dimensions increase. However, I find it curious that precisely one importantly distinct algebraic property is lost in each construction (according to this chart), and that there seems to be no immediate pattern to precisely which property is lost.

What I mean by this, is that quantifying the properties lost in each iteration yields a sequence of logic with no inherent pattern. For example, in dimensions $2^n:$ $$ \begin{align*} n=2&\implies\neg[\forall a,b\in\mathbb{H}\ \ \ \ (ab=ba)]\\ n=3&\implies\neg[\forall a,b,c\in\mathbb{O}(a(bc)=(ab)c)]\\ n=4&\implies\neg[\forall a,b\in\mathbb{S}\ \ \ \ \ (a(ab)=(aa)b)]. \end{align*}$$

Sure, one immediate pattern is that these properties have to do with multiplication, but for $n=1$ or $n>5$, it is not clear. However, perhaps even in such cases, the properties in question and be equivalently expressed by some property of multiplication. So my question is:

Is there some formula (probably second order) which describes precisely which algebraic property is lost in the $2^n$ dimensional construction? Such formula may not have to describe exactly the same properties as the one in the above table, but would yield something equisatisfiable.

Graviton
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    You should add the source for the picture. – Dietrich Burde Apr 16 '22 at 08:27
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    It's an interesting topic which, as you mention, has been asked about several times. I would just like to comment that the first column of the table, the "ordered" property, always feels completely arbitrary to me (in addition to the fact that it is generally not a property but a structure, though in the case of $\mathbb{R}$ the order is indeed unique). You can absolutely start the process with a field that does not admit any order. It just feels like people needed to find something to write in that column. – Captain Lama Apr 16 '22 at 09:58
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    Agreed with @CaptainLama. Also, as pointed out in the links, one good idea might be to separate what part of this Cayley-Dickson stuff works exactly analogously over a general field $F$ versus what depends on the reals. The crucial "arithmetic" property of $\mathbb R$ that usually does not generalize to other fields is the fact that every positive real is a square; that the multiplicative group $(\mathbb R^*, \cdot) \simeq \mathbb Z/2 \times (\mathbb R^+, \cdot)$ and further that second factor (the positive reals under multiplication) is isomorphic to the additive group $\mathbb R$. – Torsten Schoeneberg Apr 16 '22 at 19:07

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