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linear algebra – Extending the Cayley-Dickson foundation multiplication desk to uncountably many foundation parts

I’m seeking to discover an algebra with foundation parts $e_i forall iin mathbb{R}$ which, for $i in mathbb{N}$, has a multiplication desk equal to the Cayley-Dickson algebra’s multiplication desk, which has ties to the analytic XOR.

For instance, the clique of solely ${e_0, e_1, e_2, e_3}$ would breathe precisely the quaternions’ foundation parts with the identical multiplication duty because the quaternions.

The multiplication between two foundation vectors can outlined utilizing features as $$e_i e_j=(-1)^{g(i, j)} e_{f(i, j)}.$$

I’m searching for a development the place each $f$ and $g$ should breathe steady.

Is there a route of proving or disproving the actuality of this figure? There are actually areas with uncountably innumerable foundation parts. This is utilizing ZFC.

Furthermore, is it workable for the features $f$ and $g$ to breathe infinitely differentiable?

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