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I'm trying to understand which identities hold for all numbers constructed by the Cayley-Dickson construction (therefore beyond Octonions). I read that you get power-associativity $xx\cdots x$ and the flexible associativity $xyx$. Are there any others?

Now, empirically (with random elements) I found that $$ \begin{align*} x^n(x^my)&=x^m(x^ny)\\ (yx^n)x^m&=(yx^m)x^n\\ (x^ny)x^m&=x^n(yx^m)=x^nyx^m\\ \{x,y,z\}&=-\{z,y,x\} \end{align*} $$ with the associator $\{x,y,z\}=(xy)z-x(yz)$. I think the last one follows simply from flexibility in $\{x+z,y,x+z\}=0$

One could probably prove that through the construction and induction.

Are those valid identities for all Cayley-Dickson numbers? Do these identities have a name or do they derive from something known?

Are there more? (which cannot be derived from those already)

Gere
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1 Answers1

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Each $2^d$-nion $x$ satisfies a quadratic $x^2 = ax + b$, where $a = x + \overline x$ and $b = -\overline xx$ are real numbers that associate and commute with everything, so we can iteratively reduce $x^n$ to a real combination of $x$ and $1$:

$$x^n = \begin{bmatrix}a & 1 \\ b & 0\end{bmatrix}^n\begin{bmatrix}x \\ 1\end{bmatrix} = a_nx + b_n.$$

In this way, the identities about powers follow from distributivity and flexibility:

$$x^n(x^my) = a_na_mx(xy) + (a_nb_m + b_na_m)xy + b_nb_my = x^m(x^ny), \\ (yx^n)x^m = a_na_m(yx)x + (a_nb_m + b_na_m)yx + b_nb_my = (yx^m)x^n, \\ (x^ny)x^m = a_na_mxyx + a_nb_mxy + b_na_myx + b_nb_my = x^n(yx^m).$$

We can use this technique to build more general identities, such as

$$(x^my^p)(y^qx^n) = (x^my^q)(y^px^n).$$

Anders Kaseorg
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