**Update:** "The eight Cayley-Dickson doubling products", Adv. Appl. Clifford Algebras 26 (2016) pp 529–551, doi:10.1007/s00006-015-0638-6, arXiv:1707.07318 I now find that 4 of those 8 should also be discarded: each allows zero divisors at the eight dimensional stage. The 4 remaining products are denoted in the paper as $P_0$, $P_3$ (the standard doubling product), $P_4$, $P_7$. Those four produce algebras isomorphic to the standard Cayley-Dickson algebras.

I can now provide you with an easier answer. As in the first answer we use the convention that $e_pe_q=\pm e_{p\oplus q}$ where for non-negative integers $p$ and $q$, $p\oplus q$ is the bitwise exclusive 'or' of the binary representations of $p$ and $q$. Of the eight equivalent Cayley-Dickson doubling products there is one which, to my knowledge, has not been used previously by researchers.

**[edit: unfortunately, this particular doubling product cannot be classified as a Cayley-Dickson doubling product, since it produces zero divisors at the third doubling (the octonions) rather than at the fourth (the sedenions).]**

$$ (a,b)(c,d)=(ac-b^*d,da^*+bc) $$

As with the other seven doubling products we have $e_0e_p=e_pe_0=e_p$ for all $p$, $e_p^2=-1$ for $p>0$ and $e_pe_q=-e_qe_p$ for $0\ne p\ne q\ne 0$. But in addition we have the wonderful formulas

$$ \text{If } 2^N\le p<q<2^{N+1} \text{ then } e_pe_q=e_{p\oplus q} $$
and
$$ \text{If } 2^N\le p<2^{N+1}\le q \text{ then } e_pe_q=(-1)^{⌊q/2^N⌋}e_{p\oplus q} $$

where $⌊\bullet ⌋$ is the floor function.

For example, compute $e_{21}e_{5}$. First, $e_{21}e_{5}=-e_{5}e_{21}$. And since $4\le5<8<21$ we have $e_{21}e_{5}=-e_{5}e_{21}=-(-1)^{⌊21/4⌋}e_{5\oplus 21}=-(-1)^5e_{16}=e_{16}$.

Here is a a picture of the twist (or sign values for the basis vector multiplication table for the 1024-ions with a black pixel denoting a $+1$ and a white pixel denoting a $-1$. The table is indistinguishable to the naked eye for higher dimensional n-ions.

For more information see my preprint http://arxiv.org/abs/1602.02317 which is currently being refereed for publication.