Quaternions

Quaternion multiplication table:
$\begin{array}{c|cccc} & e_0 & e_1 & e_2 & e_3 \\ \hline e_0 & e_0 & e_1 & e_2 & e_3 \\ e_1 & e_1 & -e_0 & e_3 & -e_2 \\ e_2 & e_2 & -e_3 & -e_0 & e_1 \\ e_3 & e_3 & e_2 & -e_1 & -e_0 \\ \end{array}$

Quaternion multiplication can be represented using the tuple of $(e_1, e_2, e_3)$ using the following rule:
$e_a \cdot e_b = \epsilon_{abc} e_c + \delta_{ab} e_0$.

Written out individually, these rules are:
$e_1 \times e_2 = e_3 = -e_2 \times e_1$
$e_2 \times e_3 = e_1$
$e_3 \times e_1 = e_2$

$\begin{matrix} e_1 & \times e_2 & = e_3 & & \\ & e_2 & \times e_3 & = e_1 & \\ & & e_3 & \times e_1 & = e_2 \\ \end{matrix}$

Octonions

Octonion multiplication table:
$\begin{array}{c|cccccccc} & e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ \hline e_0 & e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ e_1 & e_1 & -e_0 & e_3 & -e_2 & e_5 & -e_4 & -e_7 & e_6 \\ e_2 & e_2 & -e_3 & -e_0 & e_1 & e_6 & e_7 & -e_4 & -e_5 \\ e_3 & e_3 & e_2 & -e_1 & -e_0 & e_7 & -e_6 & e_5 & -e_4 \\ e_4 & e_4 & -e_5 & -e_6 & -e_7 & -e_0 & e_1 & e_2 & e_3 \\ e_5 & e_5 & e_4 & -e_7 & e_6 & -e_1 & -e_0 & -e_3 & e_2 \\ e_6 & e_6 & e_7 & e_4 & -e_5 & -e_2 & e_3 & -e_0 & -e_1 \\ e_7 & e_7 & -e_6 & e_5 & e_4 & -e_3 & -e_2 & e_1 & -e_0 \\ \end{array}$

Within Octonion multiplication table you can find seven quaternion multiplication tables.

Let $q_i$ be the i'th quaternion multiplication table $q_i = (q_{ij}), 1 \le j \le 3$ such that the elements of $q_i$ behave as a quaternion multiplication tuple, i.e. $q_{ia} \cdot q_{ib} = \epsilon_{abc} q_{ic} + \delta_{ab} e_0$.

These seven quaternion multiplication tables of the octonions are:
$\begin{matrix} q_1 = (e_1, e_2, e_3) \\ q_2 = (e_1, e_4, e_5) \\ q_3 = (e_1, e_7, e_6) \\ q_4 = (e_2, e_4, e_6) \\ q_5 = (e_2, e_5, e_7) \\ q_6 = (e_3, e_4, e_7) \\ q_7 = (e_3, e_6, e_5) \\ \end{matrix}$

Each element is found in exactly three unique quaternion tuple.
$\begin{matrix} e_1 \in q_1, q_2, q_3 \\ e_2 \in q_1, q_4, q_5 \\ e_3 \in q_1, q_7, q_6 \\ e_4 \in q_2, q_4, q_6 \\ e_5 \in q_2, q_5, q_7 \\ e_6 \in q_3, q_4, q_7 \\ e_7 \in q_3, q_6, q_5 \\ \end{matrix}$

We can see a 1-1 relation between the tuple of quaternions that an element is contained by, and the tuple of elements that makes up a quaternion:
Define the quaternion-tuple duals of the basis elements as $\star e_i = q_i$:
$\begin{matrix} \star e_1 = q_1 = (e_1, e_2, e_3) \\ \star e_2 = q_2 = (e_1, e_4, e_5) \\ \star e_3 = q_3 = (e_1, e_7, e_6) \\ \star e_4 = q_4 = (e_2, e_4, e_6) \\ \star e_5 = q_5 = (e_2, e_5, e_7) \\ \star e_6 = q_6 = (e_3, e_4, e_7) \\ \star e_7 = q_7 = (e_3, e_6, e_5) \\ \end{matrix}$

Is there another indexing of the quternion-tuples such that elements still match the quternion-tuples, except with no element existing in its identically indexed quternion-tuple?
Just like how dual spaces work in exterior algebra?
How about looking at the triangle form of the octonion multiplication table, with $e_4 = l, e_5 = il, e_6 = jl, e_7 = kl$.
Then visually we see opposites:
$\begin{matrix} \star e_1 = q_1 = (e_1, e_4, e_5) \\ \star e_2 = q_2 = (e_2, e_4, e_6) \\ \star e_3 = q_3 = (e_3, e_4, e_7) \\ \star e_4 = q_4 = (e_1, e_2, e_3) \\ \star e_5 = q_5 = (e_1, e_7, e_6) \\ \star e_6 = q_6 = (e_2, e_5, e_7) \\ \star e_7 = q_7 = (e_3, e_6, e_5) \\ \end{matrix}$
From here can we relabel quternion-tuples to get a 1-1 matching between quternion-tuples shared by elements and elements shared by quternion-tuples, like before?
Nope.
In fact, you get one or the other.

I assigned indexes to these seven quaternions by upper-triangular first-come first-serve indexing, i.e. searching $1 \le i \lt j \le 7$, searching i ascending, then j ascending, and assigning indexes to quaternions in the order they are found. It just so happens that this indexing order possess some interesting properties. Sadly this indexing scheme does not scale. The same search rule will not assign the same quaternion indexes when searching higher Cayley-Dickson constructs. If I were to search lower-triangular indexes in the same manner then I would lose the property that $\star e_i = q_i$.
But using upper-tringular once again provides a default 1:1 index matching in sedenions of octonions to basis elements, in the same way it provides a 1:1 in octonions of quaternions to basis elements. And in sedenions, each basis element will appear in a quaternion 7 times, while quaternions can only hold 3 basis elements, so maybe there is no point to trying to reproduce the quaternion indexing within octonions when applying quaternion indexing within sedenions. Maybe upper-triangular indexing is best.
You can have the $q_i$'s and $e_i$'s quternion-tuples match, or you can have the $q_i$'s and $e_i$'s exclude the respective $q_i$ and $e_i$, but no possible permutation produces both.
Here's the 28 permutations such that the $q_i$ and $e_i$ quternion-tuples match:
$\begin{matrix} 1,2,3,4,5,6,7 \\ 1,2,3,5,4,7,6 \\ 1,3,2,6,7,5,4 \\ 1,3,2,7,6,4,5 \\ 2,4,6,1,3,5,7 \\ 2,5,7,3,1,6,4 \\ 2,6,4,1,3,7,5 \\ 2,7,5,3,1,4,6 \\ 3,4,7,5,6,1,2 \\ 3,5,6,7,4,2,1 \\ 3,6,5,4,7,2,1 \\ 3,7,4,6,5,1,2 \\ 4,1,5,2,6,3,7 \\ 4,1,5,3,7,2,6 \\ 4,2,6,1,5,3,7 \\ 4,3,7,2,6,1,5 \\ 5,1,4,6,3,7,2 \\ 5,1,4,7,2,6,3 \\ 5,2,7,4,1,6,3 \\ 5,3,6,7,2,4,1 \\ 6,4,2,1,7,5,3 \\ 6,5,3,2,4,7,1 \\ 6,7,1,2,4,5,3 \\ 6,7,1,3,5,4,2 \\ 7,4,3,5,2,1,6 \\ 7,5,2,6,1,3,4 \\ 7,6,1,4,3,2,5 \\ 7,6,1,5,2,3,4 \\ \end{matrix}$
Here's the 144 permutations such that either the $e_i$'s don't contain $q_i$ or the $q_i$'s don't contain $e_i$:
$\begin{matrix} 4,2,3,1,6,5,7 \\ 4,2,3,5,1,6,7 \\ 4,2,3,5,6,1,7 \\ 4,2,3,7,6,5,1 \\ 4,2,5,1,3,6,7 \\ 4,2,5,3,1,6,7 \\ 4,2,5,3,6,1,7 \\ 4,2,5,7,3,6,1 \\ 4,3,2,1,6,5,7 \\ 4,3,2,5,1,6,7 \\ 4,3,2,5,6,1,7 \\ 4,3,2,7,6,5,1 \\ 4,3,5,1,6,2,7 \\ 4,3,5,7,1,6,2 \\ 4,3,5,7,6,1,2 \\ 4,3,5,7,6,2,1 \\ 4,6,2,1,3,5,7 \\ 4,6,2,3,1,5,7 \\ 4,6,2,5,3,1,7 \\ 4,6,2,7,3,5,1 \\ 4,6,3,5,1,2,7 \\ 4,6,3,7,1,5,2 \\ 4,6,5,1,3,2,7 \\ 4,6,5,3,1,2,7 \\ 4,6,5,7,3,1,2 \\ 4,6,5,7,3,2,1 \\ 4,7,2,3,6,5,1 \\ 4,7,2,5,3,6,1 \\ 4,7,3,1,6,5,2 \\ 4,7,3,5,1,6,2 \\ 4,7,3,5,6,1,2 \\ 4,7,3,5,6,2,1 \\ 4,7,5,1,3,6,2 \\ 4,7,5,3,1,6,2 \\ 4,7,5,3,6,1,2 \\ 4,7,5,3,6,2,1 \\ 5,2,3,1,4,6,7 \\ 5,2,3,7,1,6,4 \\ 5,2,3,7,4,6,1 \\ 5,2,3,7,6,1,4 \\ 5,2,4,1,3,6,7 \\ 5,2,4,3,1,6,7 \\ 5,2,4,3,6,1,7 \\ 5,2,4,7,3,6,1 \\ 5,3,2,1,4,6,7 \\ 5,3,2,7,1,6,4 \\ 5,3,2,7,4,6,1 \\ 5,3,2,7,6,1,4 \\ 5,3,4,1,6,2,7 \\ 5,3,4,7,1,6,2 \\ 5,3,4,7,6,1,2 \\ 5,3,4,7,6,2,1 \\ 5,6,2,3,4,1,7 \\ 5,6,2,7,3,1,4 \\ 5,6,3,1,4,2,7 \\ 5,6,3,7,1,2,4 \\ 5,6,3,7,4,1,2 \\ 5,6,3,7,4,2,1 \\ 5,6,4,1,3,2,7 \\ 5,6,4,3,1,2,7 \\ 5,6,4,7,3,1,2 \\ 5,6,4,7,3,2,1 \\ 5,7,2,1,3,6,4 \\ 5,7,2,3,1,6,4 \\ 5,7,2,3,4,6,1 \\ 5,7,2,3,6,1,4 \\ 5,7,3,1,4,6,2 \\ 5,7,3,1,6,2,4 \\ 5,7,4,1,3,6,2 \\ 5,7,4,3,1,6,2 \\ 5,7,4,3,6,1,2 \\ 5,7,4,3,6,2,1 \\ 6,2,3,1,4,5,7 \\ 6,2,3,5,4,1,7 \\ 6,2,3,7,1,5,4 \\ 6,2,3,7,4,5,1 \\ 6,2,4,1,3,5,7 \\ 6,2,4,3,1,5,7 \\ 6,2,4,5,3,1,7 \\ 6,2,4,7,3,5,1 \\ 6,2,5,3,4,1,7 \\ 6,2,5,7,3,1,4 \\ 6,3,2,1,4,5,7 \\ 6,3,2,5,4,1,7 \\ 6,3,2,7,1,5,4 \\ 6,3,2,7,4,5,1 \\ 6,3,4,5,1,2,7 \\ 6,3,4,7,1,5,2 \\ 6,3,5,1,4,2,7 \\ 6,3,5,7,1,2,4 \\ 6,3,5,7,4,1,2 \\ 6,3,5,7,4,2,1 \\ 6,7,2,1,3,5,4 \\ 6,7,2,3,1,5,4 \\ 6,7,2,3,4,5,1 \\ 6,7,2,5,3,1,4 \\ 6,7,3,1,4,5,2 \\ 6,7,3,5,1,2,4 \\ 6,7,3,5,4,1,2 \\ 6,7,3,5,4,2,1 \\ 6,7,4,1,3,5,2 \\ 6,7,4,3,1,5,2 \\ 6,7,4,5,3,1,2 \\ 6,7,4,5,3,2,1 \\ 6,7,5,1,3,2,4 \\ 6,7,5,3,1,2,4 \\ 6,7,5,3,4,1,2 \\ 6,7,5,3,4,2,1 \\ 7,2,3,1,6,5,4 \\ 7,2,3,5,1,6,4 \\ 7,2,3,5,4,6,1 \\ 7,2,3,5,6,1,4 \\ 7,2,4,3,6,5,1 \\ 7,2,4,5,3,6,1 \\ 7,2,5,1,3,6,4 \\ 7,2,5,3,1,6,4 \\ 7,2,5,3,4,6,1 \\ 7,2,5,3,6,1,4 \\ 7,3,2,1,6,5,4 \\ 7,3,2,5,1,6,4 \\ 7,3,2,5,4,6,1 \\ 7,3,2,5,6,1,4 \\ 7,3,4,1,6,5,2 \\ 7,3,4,5,1,6,2 \\ 7,3,4,5,6,1,2 \\ 7,3,4,5,6,2,1 \\ 7,3,5,1,4,6,2 \\ 7,3,5,1,6,2,4 \\ 7,6,2,1,3,5,4 \\ 7,6,2,3,1,5,4 \\ 7,6,2,3,4,5,1 \\ 7,6,2,5,3,1,4 \\ 7,6,3,1,4,5,2 \\ 7,6,3,5,1,2,4 \\ 7,6,3,5,4,1,2 \\ 7,6,3,5,4,2,1 \\ 7,6,4,1,3,5,2 \\ 7,6,4,3,1,5,2 \\ 7,6,4,5,3,1,2 \\ 7,6,4,5,3,2,1 \\ 7,6,5,1,3,2,4 \\ 7,6,5,3,1,2,4 \\ 7,6,5,3,4,1,2 \\ 7,6,5,3,4,2,1 \\ \end{matrix}$
...and the list of the rest of the permutations -- that neither match $e_i$'s with $q_i$'s nor exclude either $e_i$'s in $q_i$'s or $q_i$'s in $e_i$'s -- is much much longer...
In fact it is 4868 permutations long.
One thing to note: for any permutation, no $e_i$ contains $q_i$ is true if and only if no $q_i$ contains $e_i$.

There is a unique sequence of 7 elements that can simultaneously describe all 7 quternion multiplication tuples within the octonions.
We can combine all 7 quternion-tuples using this fact, to form a multiplication on a Mobius strip.
We will specify our Mobius strip with the rule that three elements of a particular spacing will form a quternion-tuple.

To find a particular spacing that works, we will create an adjacency matrix of the quternion-tuples for any particular spacing.
This is an intuitive process for representing all quternion-tuples within an octonion multiplication table. I am describing it here because the same process will be used later with more complex embeddings.
To form this matrix, start with a vector formed from 1's where quternion-tuple elements would go and 0's otherwise.
For example, for the 1st, 2nd, 3rd we get $[1, 1, 1, 0, 0, 0, 0]$
Next, rotate the elements of the vector around, to get this:
$M = \left[\begin{matrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ \end{matrix}\right]$
Multiplying this matrix by its transpose shows the number of elements that each pair of quternion-tuples have in common:
$M \cdot M^T = \left[\begin{matrix} 3 & 2 & 1 & 0 & 0 & 1 & 2 \\ 2 & 3 & 2 & 1 & 0 & 0 & 1 \\ 1 & 2 & 3 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 3 & 2 & 1 & 0 \\ 0 & 0 & 1 & 2 & 3 & 2 & 1 \\ 1 & 0 & 0 & 1 & 2 & 3 & 2 \\ 2 & 1 & 0 & 0 & 1 & 2 & 3 \\ \end{matrix}\right]$

This shows that the arrangement of placing at the 1st, 2nd, 3rd does not arrange quternion-tuples evenly.
We're looking for even distribution of elements, such that $(M \cdot M^T)_{ij}$ is equal for all $i \ne j$.

Try again with spacings 1st, 2nd, 4th.
Rotate elements $[1, 1, 0, 1, 0, 0, 0]$
$M = \left[\begin{matrix} 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ \end{matrix}\right]$
$M \cdot M^T = \left[\begin{matrix} 3 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 3 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 3 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 3 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 3 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 3 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 3 \\ \end{matrix}\right]$

This arrangement, however, does distribute elements evenly.

We find following spacings do distribute elements evenly:
(1st, 2nd, 4th) = primary ordering
(1st, 3rd, 7th) = primary ordering x 2 mod 7
(1st, 3rd, 4th) = primary ordering x 3 mod 7
(1st, 5th, 6th) = primary ordering x 4 mod 7
(1st, 2nd, 6th) = primary ordering x 5 mod 7
(1st, 5th, 7th) = primary ordering x 6 mod 7

If you examine all possible embeddings of 3 elements into 7 elements such that all elements share one neighbors in common, you find that there is only one unique isomorphism.
Each of these are isomorphisms of 1st, 2nd, 4th spacing.
So there is only one unique arrangements of quternion-tuples such that they share an equal number of elements.

Using this you can start with the first available quternion-tuple and successively add quternion-tuples that fulfill the constraint, and eventually represent the octonion multiplication table as a tuple and quternion-tuple multiplication occurring between any element, the next, and the 4th next.
For octonion-tuple elements $(e_1, e_2, e_4, e_3, e_6, -e_7, e_5)$, looking at the 1st, 2nd, and 4th, gives us we see quaternion-tuples:
$ q_1 = (e_1, e_2, e_3), q_4 = (e_2, e_4, e_6), q_6 = (e_4, e_3, -e_7), q_7 = (e_3, e_6, e_5), q_3 = (e_6, -e_7, e_1), q_5 = (-e_7, e_5, e_2), q_2 = (e_5, e_1, e_4), $.
If you represent these quternion-tuples as triangles on a mesh then you come up with a Mobius strip.

For entertainment, here are all octonion-tuples formed from this 1st, 2nd, 4th spacing.
I.e. all enumerations of 7 elements such that, starting at any element, picking that element, the next, and the 4th-next produces a quaternion multiplication quternion-tuple.
$\begin{matrix} ( & e_1 & e_2 & e_4 & e_3 & e_6 &-e_7 & e_5 & ) \\ ( & e_1 & e_2 & e_5 & e_3 & e_7 & e_6 &-e_4 & ) \\ ( & e_1 & e_4 & e_6 & e_5 & e_2 & e_3 &-e_7 & ) \\ ( & e_1 & e_4 & e_7 & e_5 & e_3 &-e_2 & e_6 & ) \\ ( & e_1 & e_7 & e_2 & e_6 & e_5 &-e_4 & e_3 & ) \\ ( & e_1 & e_7 & e_3 & e_6 & e_4 & e_5 &-e_2 & ) \\ ( & e_2 & e_3 & e_4 & e_1 & e_7 &-e_5 & e_6 & ) \\ ( & e_2 & e_3 & e_6 & e_1 & e_5 & e_7 &-e_4 & ) \\ ( & e_2 & e_4 & e_5 & e_6 & e_1 &-e_3 & e_7 & ) \\ ( & e_2 & e_4 & e_7 & e_6 & e_3 & e_1 &-e_5 & ) \\ ( & e_2 & e_5 & e_1 & e_7 & e_4 & e_6 &-e_3 & ) \\ ( & e_2 & e_5 & e_3 & e_7 & e_6 &-e_4 & e_1 & ) \\ ( & e_3 & e_1 & e_4 & e_2 & e_5 &-e_6 & e_7 & ) \\ ( & e_3 & e_1 & e_7 & e_2 & e_6 & e_5 &-e_4 & ) \\ ( & e_3 & e_4 & e_5 & e_7 & e_1 & e_2 &-e_6 & ) \\ ( & e_3 & e_4 & e_6 & e_7 & e_2 &-e_1 & e_5 & ) \\ ( & e_3 & e_6 & e_1 & e_5 & e_7 &-e_4 & e_2 & ) \\ ( & e_3 & e_6 & e_2 & e_5 & e_4 & e_7 &-e_1 & ) \\ ( & e_4 & e_5 & e_7 & e_1 & e_2 &-e_6 & e_3 & ) \\ ( & e_4 & e_5 & e_3 & e_1 & e_6 & e_2 &-e_7 & ) \\ ( & e_4 & e_6 & e_1 & e_2 & e_7 & e_3 &-e_5 & ) \\ ( & e_4 & e_6 & e_5 & e_2 & e_3 &-e_7 & e_1 & ) \\ ( & e_4 & e_7 & e_6 & e_3 & e_1 &-e_5 & e_2 & ) \\ ( & e_4 & e_7 & e_2 & e_3 & e_5 & e_1 &-e_6 & ) \\ ( & e_5 & e_1 & e_2 & e_4 & e_3 & e_6 &-e_7 & ) \\ ( & e_5 & e_1 & e_7 & e_4 & e_6 &-e_3 & e_2 & ) \\ ( & e_5 & e_7 & e_6 & e_2 & e_1 & e_4 &-e_3 & ) \\ ( & e_5 & e_7 & e_3 & e_2 & e_4 &-e_1 & e_6 & ) \\ ( & e_5 & e_3 & e_1 & e_6 & e_2 &-e_7 & e_4 & ) \\ ( & e_5 & e_3 & e_4 & e_6 & e_7 & e_2 &-e_1 & ) \\ ( & e_6 & e_1 & e_2 & e_7 & e_3 &-e_5 & e_4 & ) \\ ( & e_6 & e_1 & e_4 & e_7 & e_5 & e_3 &-e_2 & ) \\ ( & e_6 & e_2 & e_3 & e_4 & e_1 & e_7 &-e_5 & ) \\ ( & e_6 & e_2 & e_5 & e_4 & e_7 &-e_1 & e_3 & ) \\ ( & e_6 & e_5 & e_1 & e_3 & e_4 &-e_2 & e_7 & ) \\ ( & e_6 & e_5 & e_7 & e_3 & e_2 & e_4 &-e_1 & ) \\ ( & e_7 & e_6 & e_2 & e_1 & e_4 &-e_3 & e_5 & ) \\ ( & e_7 & e_6 & e_5 & e_1 & e_3 & e_4 &-e_2 & ) \\ ( & e_7 & e_2 & e_3 & e_5 & e_1 &-e_6 & e_4 & ) \\ ( & e_7 & e_2 & e_4 & e_5 & e_6 & e_1 &-e_3 & ) \\ ( & e_7 & e_3 & e_1 & e_4 & e_2 & e_5 &-e_6 & ) \\ ( & e_7 & e_3 & e_6 & e_4 & e_5 &-e_2 & e_1 & ) \\ \end{matrix}$

All represent the same octonion multiplication table, so the first is as good a representation as any:
$(e_1, e_2, e_4, e_3, e_6, -e_7, e_5)$

$\begin{matrix} e_1 & \times e_2 & & = e_3 & & & & \\ & e_2 & \times e_4 & & = e_6 & & & \\ & & e_4 & \times e_3 & & = -e_7 & & \\ & & & e_3 & \times e_6 & & = e_5 \\ = e_1 & & & & e_6 & \times -e_7 & & \\ & = e_2 & & & & -e_7 & \times e_5 \\ \times e_1 & & = e_4 & & & & e_5 \\ \end{matrix}$

Sedenions

Sedenion multiplication is 15 imaginary dimensions.

Sedenion multiplication table.
$\begin{array}{c|cccccccccccccccc} & e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 & e_8 & e_9 & e_{10} & e_{11} & e_{12} & e_{13} & e_{14} & e_{15} \\ \hline e_0 & e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 & e_8 & e_9 & e_{10} & e_{11} & e_{12} & e_{13} & e_{14} & e_{15} \\ e_1 & e_1 & -e_0 & e_3 & -e_2 & e_5 & -e_4 & -e_7 & e_6 & e_9 & -e_8 & -e_{11} & e_{10} & -e_{13} & e_{12} & e_{15} & -e_{14} \\ e_2 & e_2 & -e_3 & -e_0 & e_1 & e_6 & e_7 & -e_4 & -e_5 & e_{10} & e_{11} & -e_8 & -e_9 & -e_{14} & -e_{15} & e_{12} & e_{13} \\ e_3 & e_3 & e_2 & -e_1 & -e_0 & e_7 & -e_6 & e_5 & -e_4 & e_{11} & -e_{10} & e_9 & -e_8 & -e_{15} & e_{14} & -e_{13} & e_{12} \\ e_4 & e_4 & -e_5 & -e_6 & -e_7 & -e_0 & e_1 & e_2 & e_3 & e_{12} & e_{13} & e_{14} & e_{15} & -e_8 & -e_9 & -e_{10} & -e_{11} \\ e_5 & e_5 & e_4 & -e_7 & e_6 & -e_1 & -e_0 & -e_3 & e_2 & e_{13} & -e_{12} & e_{15} & -e_{14} & e_9 & -e_8 & e_{11} & -e_{10} \\ e_6 & e_6 & e_7 & e_4 & -e_5 & -e_2 & e_3 & -e_0 & -e_1 & e_{14} & -e_{15} & -e_{12} & e_{13} & e_{10} & -e_{11} & -e_8 & e_9 \\ e_7 & e_7 & -e_6 & e_5 & e_4 & -e_3 & -e_2 & e_1 & -e_0 & e_{15} & e_{14} & -e_{13} & -e_{12} & e_{11} & e_{10} & -e_9 & -e_8 \\ e_8 & e_8 & -e_9 & -e_{10} & -e_{11} & -e_{12} & -e_{13} & -e_{14} & -e_{15} & -e_0 & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 \\ e_9 & e_9 & e_8 & -e_{11} & e_{10} & -e_{13} & e_{12} & e_{15} & -e_{14} & -e_1 & -e_0 & -e_3 & e_2 & -e_5 & e_4 & e_7 & -e_6 \\ e_{10} & e_{10} & e_{11} & e_8 & -e_9 & -e_{14} & -e_{15} & e_{12} & e_{13} & -e_2 & e_3 & -e_0 & -e_1 & -e_6 & -e_7 & e_4 & e_5 \\ e_{11} & e_{11} & -e_{10} & e_9 & e_8 & -e_{15} & e_{14} & -e_{13} & e_{12} & -e_3 & -e_2 & e_1 & -e_0 & -e_7 & e_6 & -e_5 & e_4 \\ e_{12} & e_{12} & e_{13} & e_{14} & e_{15} & e_8 & -e_9 & -e_{10} & -e_{11} & -e_4 & e_5 & e_6 & e_7 & -e_0 & -e_1 & -e_2 & -e_3 \\ e_{13} & e_{13} & -e_{12} & e_{15} & -e_{14} & e_9 & e_8 & e_{11} & -e_{10} & -e_5 & -e_4 & e_7 & -e_6 & e_1 & -e_0 & e_3 & -e_2 \\ e_{14} & e_{14} & -e_{15} & -e_{12} & e_{13} & e_{10} & -e_{11} & e_8 & e_9 & -e_6 & -e_7 & -e_4 & e_5 & e_2 & -e_3 & -e_0 & e_1 \\ e_{15} & e_{15} & e_{14} & -e_{13} & -e_{12} & e_{11} & e_{10} & -e_9 & e_8 & -e_7 & e_6 & -e_5 & -e_4 & e_3 & e_2 & -e_1 & -e_0 \\ \end{array}$

Table of which quternion-tuple each pair of elements belong to:
$\begin{array}{c|cccccccccccccccc} & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 & e_8 & e_9 & e_{10} & e_{11} & e_{12} & e_{13} & e_{14} & e_{15} \\ \hline e_1 & & q_1=(e_1,e_2,e_3) & q_1=(e_1,e_2,e_3) & q_2=(e_1,e_4,e_5) & q_2=(e_1,e_4,e_5) & q_3=(e_1,e_7,e_6) & q_3=(e_1,e_7,e_6) & (e_1,e_8,e_9) & (e_1,e_8,e_9) & (e_1,e_{11},e_{10}) & (e_1,e_{11},e_{10}) & (e_1,e_{13},e_{12}) & (e_1,e_{13},e_{12}) & (e_1,e_{14},e_{15}) & (e_1,e_{14},e_{15}) \\ e_2 & q_1=(e_1,e_2,e_3) & & q_1=(e_1,e_2,e_3) & q_4=(e_2,e_4,e_6) & q_5=(e_2,e_5,e_7) & q_4=(e_2,e_4,e_6) & q_5=(e_2,e_5,e_7) & (e_2,e_8,e_{10}) & (e_2,e_9,e_{11}) & (e_2,e_8,e_{10}) & (e_2,e_9,e_{11}) & (e_2,e_{14},e_{12}) & (e_2,e_{15},e_{13}) & (e_2,e_{14},e_{12}) & (e_2,e_{15},e_{13}) \\ e_3 & q_1=(e_1,e_2,e_3) & q_1=(e_1,e_2,e_3) & & q_6=(e_3,e_4,e_7) & q_7=(e_3,e_6,e_5) & q_7=(e_3,e_6,e_5) & q_6=(e_3,e_4,e_7) & (e_3,e_8,e_{11}) & (e_3,e_{10},e_9) & (e_3,e_{10},e_9) & (e_3,e_8,e_{11}) & (e_3,e_{15},e_{12}) & (e_3,e_{13},e_{14}) & (e_3,e_{13},e_{14}) & (e_3,e_{15},e_{12}) \\ e_4 & q_2=(e_1,e_4,e_5) & q_4=(e_2,e_4,e_6) & q_6=(e_3,e_4,e_7) & & q_2=(e_1,e_4,e_5) & q_4=(e_2,e_4,e_6) & q_6=(e_3,e_4,e_7) & (e_4,e_8,e_{12}) & (e_4,e_9,e_{13}) & (e_4,e_{10},e_{14}) & (e_4,e_{11},e_{15}) & (e_4,e_8,e_{12}) & (e_4,e_9,e_{13}) & (e_4,e_{10},e_{14}) & (e_4,e_{11},e_{15}) \\ e_5 & q_2=(e_1,e_4,e_5) & q_5=(e_2,e_5,e_7) & q_7=(e_3,e_6,e_5) & q_2=(e_1,e_4,e_5) & & q_7=(e_3,e_6,e_5) & q_5=(e_2,e_5,e_7) & (e_5,e_8,e_{13}) & (e_5,e_{12},e_9) & (e_5,e_{10},e_{15}) & (e_5,e_{14},e_{11}) & (e_5,e_{12},e_9) & (e_5,e_8,e_{13}) & (e_5,e_{14},e_{11}) & (e_5,e_{10},e_{15}) \\ e_6 & q_3=(e_1,e_7,e_6) & q_4=(e_2,e_4,e_6) & q_7=(e_3,e_6,e_5) & q_4=(e_2,e_4,e_6) & q_7=(e_3,e_6,e_5) & & q_3=(e_1,e_7,e_6) & (e_6,e_8,e_{14}) & (e_6,e_{15},e_9) & (e_6,e_{12},e_{10}) & (e_6,e_{11},e_{13}) & (e_6,e_{12},e_{10}) & (e_6,e_{11},e_{13}) & (e_6,e_8,e_{14}) & (e_6,e_{15},e_9) \\ e_7 & q_3=(e_1,e_7,e_6) & q_5=(e_2,e_5,e_7) & q_6=(e_3,e_4,e_7) & q_6=(e_3,e_4,e_7) & q_5=(e_2,e_5,e_7) & q_3=(e_1,e_7,e_6) & & (e_7,e_8,e_{15}) & (e_7,e_9,e_{14}) & (e_7,e_{13},e_{10}) & (e_7,e_{12},e_{11}) & (e_7,e_{12},e_{11}) & (e_7,e_{13},e_{10}) & (e_7,e_9,e_{14}) & (e_7,e_8,e_{15}) \\ e_8 & (e_1,e_8,e_9) & (e_2,e_8,e_{10}) & (e_3,e_8,e_{11}) & (e_4,e_8,e_{12}) & (e_5,e_8,e_{13}) & (e_6,e_8,e_{14}) & (e_7,e_8,e_{15}) & & (e_1,e_8,e_9) & (e_2,e_8,e_{10}) & (e_3,e_8,e_{11}) & (e_4,e_8,e_{12}) & (e_5,e_8,e_{13}) & (e_6,e_8,e_{14}) & (e_7,e_8,e_{15}) \\ e_9 & (e_1,e_8,e_9) & (e_2,e_9,e_{11}) & (e_3,e_{10},e_9) & (e_4,e_9,e_{13}) & (e_5,e_{12},e_9) & (e_6,e_{15},e_9) & (e_7,e_9,e_{14}) & (e_1,e_8,e_9) & & (e_3,e_{10},e_9) & (e_2,e_9,e_{11}) & (e_5,e_{12},e_9) & (e_4,e_9,e_{13}) & (e_7,e_9,e_{14}) & (e_6,e_{15},e_9) \\ e_{10} & (e_1,e_{11},e_{10}) & (e_2,e_8,e_{10}) & (e_3,e_{10},e_9) & (e_4,e_{10},e_{14}) & (e_5,e_{10},e_{15}) & (e_6,e_{12},e_{10}) & (e_7,e_{13},e_{10}) & (e_2,e_8,e_{10}) & (e_3,e_{10},e_9) & & (e_1,e_{11},e_{10}) & (e_6,e_{12},e_{10}) & (e_7,e_{13},e_{10}) & (e_4,e_{10},e_{14}) & (e_5,e_{10},e_{15}) \\ e_{11} & (e_1,e_{11},e_{10}) & (e_2,e_9,e_{11}) & (e_3,e_8,e_{11}) & (e_4,e_{11},e_{15}) & (e_5,e_{14},e_{11}) & (e_6,e_{11},e_{13}) & (e_7,e_{12},e_{11}) & (e_3,e_8,e_{11}) & (e_2,e_9,e_{11}) & (e_1,e_{11},e_{10}) & & (e_7,e_{12},e_{11}) & (e_6,e_{11},e_{13}) & (e_5,e_{14},e_{11}) & (e_4,e_{11},e_{15}) \\ e_{12} & (e_1,e_{13},e_{12}) & (e_2,e_{14},e_{12}) & (e_3,e_{15},e_{12}) & (e_4,e_8,e_{12}) & (e_5,e_{12},e_9) & (e_6,e_{12},e_{10}) & (e_7,e_{12},e_{11}) & (e_4,e_8,e_{12}) & (e_5,e_{12},e_9) & (e_6,e_{12},e_{10}) & (e_7,e_{12},e_{11}) & & (e_1,e_{13},e_{12}) & (e_2,e_{14},e_{12}) & (e_3,e_{15},e_{12}) \\ e_{13} & (e_1,e_{13},e_{12}) & (e_2,e_{15},e_{13}) & (e_3,e_{13},e_{14}) & (e_4,e_9,e_{13}) & (e_5,e_8,e_{13}) & (e_6,e_{11},e_{13}) & (e_7,e_{13},e_{10}) & (e_5,e_8,e_{13}) & (e_4,e_9,e_{13}) & (e_7,e_{13},e_{10}) & (e_6,e_{11},e_{13}) & (e_1,e_{13},e_{12}) & & (e_3,e_{13},e_{14}) & (e_2,e_{15},e_{13}) \\ e_{14} & (e_1,e_{14},e_{15}) & (e_2,e_{14},e_{12}) & (e_3,e_{13},e_{14}) & (e_4,e_{10},e_{14}) & (e_5,e_{14},e_{11}) & (e_6,e_8,e_{14}) & (e_7,e_9,e_{14}) & (e_6,e_8,e_{14}) & (e_7,e_9,e_{14}) & (e_4,e_{10},e_{14}) & (e_5,e_{14},e_{11}) & (e_2,e_{14},e_{12}) & (e_3,e_{13},e_{14}) & & (e_1,e_{14},e_{15}) \\ e_{15} & (e_1,e_{14},e_{15}) & (e_2,e_{15},e_{13}) & (e_3,e_{15},e_{12}) & (e_4,e_{11},e_{15}) & (e_5,e_{10},e_{15}) & (e_6,e_{15},e_9) & (e_7,e_8,e_{15}) & (e_7,e_8,e_{15}) & (e_6,e_{15},e_9) & (e_5,e_{10},e_{15}) & (e_4,e_{11},e_{15}) & (e_3,e_{15},e_{12}) & (e_2,e_{15},e_{13}) & (e_1,e_{14},e_{15}) \\ \end{array}$

There are 35 unique quternion-tuples:
$q_1 = (e_1, e_2, e_3),$
$q_2 = (e_1, e_4, e_5),$
$q_3 = (e_1, e_7, e_6),$
$q_4 = (e_2, e_4, e_6),$
$q_5 = (e_2, e_5, e_7),$
$q_6 = (e_3, e_4, e_7),$
$q_7 = (e_3, e_6, e_5),$
$q_8 = (e_1, e_8, e_9),$
$q_9 = (e_1, e_{11}, e_{10}),$
$q_a = (e_1, e_{13}, e_{12}),$
$q_b = (e_1, e_{14}, e_{15}),$
$q_c = (e_2, e_8, e_{10}),$
$q_d = (e_2, e_9, e_{11}),$
$q_e = (e_2, e_{14}, e_{12}),$
$q_f = (e_2, e_{15}, e_{13}),$
$q_g = (e_3, e_8, e_{11}),$
$q_h = (e_3, e_{10}, e_9),$
$q_i = (e_3, e_{13}, e_{14}),$
$q_j = (e_3, e_{15}, e_{12}),$
$q_k = (e_4, e_8, e_{12}),$
$q_l = (e_4, e_9, e_{13}),$
$q_m = (e_4, e_{10}, e_{14}),$
$q_n = (e_4, e_{11}, e_{15}),$
$q_o = (e_5, e_8, e_{13}),$
$q_p = (e_5, e_{10}, e_{15}),$
$q_q = (e_5, e_{12}, e_9),$
$q_r = (e_5, e_{14}, e_{11}),$
$q_s = (e_6, e_8, e_{14}),$
$q_t = (e_6, e_{11}, e_{13}),$
$q_u = (e_6, e_{12}, e_{10}),$
$q_v = (e_6, e_{15}, e_9),$
$q_w = (e_7, e_8, e_{15}),$
$q_x = (e_7, e_9, e_{14}),$
$q_y = (e_7, e_{12}, e_{11}),$
$q_z = (e_7, e_{13}, e_{10})$

... such that ...
$e_1 \in q_1, q_2, q_3, q_8, q_9, q_a, q_b$
$e_2 \in ...$

... I'm looking for an indexing of q's so that e_i is in q_a q_b q_c and q_i contains e_a e_b e_c for matching i,a,b,c .... ... but that's not going to happen because e_i will be in seven q's, so what of the other 4?
... another way to maintain indexes between different sized Cayley-Dickson constructs is to recursively search for lower C.D.'s in-order, i.e. don't serach for quaternions in sedenions, instead search for octonions in sedenions and then search for quaternions in octonions. This will produce a different ordering.
But if we do use an upper-triangular first-come ordering to quaternions within sedenions then the octonions containing basis elements wont match the octonions containing quaternion indexes ... unless the first 7 quaternions does match with the 7 quaternions of the Cayley-Dickson-3 octonion construction.
Indexing quaternions by upper-triangular cycling all pairs of basis elements:
$q_1 = (e_1, e_2, e_3)$
$q_2 = (e_1, e_4, e_5)$
$q_3 = (e_1, e_6, e_7)$
$q_4 = (e_1, e_8, e_9)$
$q_5 = (e_1, e_{10}, e_{11})$
$q_6 = (e_1, e_{12}, e_{13})$
$q_7 = (e_1, e_{14}, e_{15})$
$q_8 = (e_2, e_4, e_6)$
$q_9 = (e_2, e_5, e_7)$
$q_{10} = (e_2, e_8, e_{10})$
$q_{11} = (e_2, e_9, e_{11})$
$q_{12} = (e_2, e_{12}, e_{14})$
$q_{13} = (e_2, e_{13}, e_{15})$
$q_{14} = (e_3, e_4, e_7)$
$q_{15} = (e_3, e_5, e_6)$
$q_{16} = (e_3, e_8, e_{11})$
$q_{17} = (e_3, e_9, e_{10})$
$q_{18} = (e_3, e_{12}, e_{15})$
$q_{19} = (e_3, e_{13}, e_{14})$
$q_{20} = (e_4, e_8, e_{12})$
$q_{21} = (e_4, e_9, e_{13})$
$q_{22} = (e_4, e_{10}, e_{14})$
$q_{23} = (e_4, e_{11}, e_{15})$
$q_{24} = (e_5, e_8, e_{13})$
$q_{25} = (e_5, e_9, e_{12})$
$q_{26} = (e_5, e_{10}, e_{15})$
$q_{27} = (e_5, e_{11}, e_{14})$
$q_{28} = (e_6, e_8, e_{14})$
$q_{29} = (e_6, e_9, e_{15})$
$q_{30} = (e_6, e_{10}, e_{12})$
$q_{31} = (e_6, e_{11}, e_{13})$
$q_{32} = (e_7, e_8, e_{15})$
$q_{33} = (e_7, e_9, e_{14})$
$q_{34} = (e_7, e_{10}, e_{13})$
$q_{35} = (e_7, e_{11}, e_{12})$

Indexing quaternions by upper-triangular searching for octonions, then octonions searching quaternions:
$q_1 = (e_1, e_2, e_3)$
$q_2 = (e_1, e_4, e_5)$
$q_3 = (e_1, e_6, e_7)$
$q_4 = (e_2, e_4, e_6)$
$q_5 = (e_2, e_5, e_7)$
$q_6 = (e_3, e_4, e_7)$
$q_7 = (e_3, e_5, e_6)$
$q_8 = (e_1, e_8, e_9)$
$q_9 = (e_1, e_{10}, e_{11})$
$q_{10} = (e_2, e_8, e_{10})$
$q_{11} = (e_2, e_9, e_{11})$
$q_{12} = (e_3, e_8, e_{11})$
$q_{13} = (e_3, e_9, e_{10})$
$q_{14} = (e_1, e_{12}, e_{13})$
$q_{15} = (e_1, e_{14}, e_{15})$
$q_{16} = (e_2, e_{12}, e_{14})$
$q_{17} = (e_2, e_{13}, e_{15})$
$q_{18} = (e_3, e_{12}, e_{15})$
$q_{19} = (e_3, e_{13}, e_{14})$
$q_{20} = (e_4, e_8, e_{12})$
$q_{21} = (e_4, e_9, e_{13})$
$q_{22} = (e_5, e_8, e_{13})$
$q_{23} = (e_5, e_9, e_{12})$
$q_{24} = (e_4, e_{10}, e_{14})$
$q_{25} = (e_4, e_{11}, e_{15})$
$q_{26} = (e_5, e_{10}, e_{15})$
$q_{27} = (e_5, e_{11}, e_{14})$
$q_{28} = (e_6, e_8, e_{14})$
$q_{29} = (e_6, e_9, e_{15})$
$q_{30} = (e_7, e_8, e_{15})$
$q_{31} = (e_7, e_9, e_{14})$
$q_{32} = (e_6, e_{10}, e_{12})$
$q_{33} = (e_6, e_{11}, e_{13})$
$q_{34} = (e_7, e_{10}, e_{13})$
$q_{35} = (e_7, e_{11}, e_{12})$

There are 15 tuples representing octonion multiplication tables found within the sedenions:
$o_1 = ( e_1, e_2, e_4, e_3, e_6, -e_7, e_5 )$ is made of quaternions $(q_1, q_4, q_6, q_7, q_3, q_5, q_2)$.
$o_2 = ( e_1, e_2, e_8, e_3, e_{10}, -e_{11}, e_9 )$
$o_3 = ( e_1, e_2, e_{14}, e_3, e_{12}, e_{13}, -e_{15}, -e_1, e_2, e_{14}, -e_3, e_{12}, -e_{13}, e_{15} )$ with $(e_2, e_{14}, e_{12})$ preserving signs and $(e_1, e_3, e_{13}, -e_{15})$ alternating signs.
$o_4 = ( e_1, e_4, e_8, e_5, e_{12}, -e_{13}, e_9 )$
$o_5 = ( e_1, e_4, e_{10}, e_5, e_{14}, -e_{15}, e_{11}, -e_1, e_4, e_{10}, -e_5, e_{14}, e_{15}, -e_{11} )$ with $(e_4, e_{10}, e_{14})$ preserving signs and $(e_1, e_5, -e_{15}, e_{11})$ alternating signs.
$o_6 = ( e_1, e_7, e_8, e_6, e_{15}, -e_{14}, e_9 )$
$o_7 = ( e_1, e_7, e_{12}, e_6, e_{11}, -e_{10}, e_{13}, -e_1, e_7, e_{12}, -e_6, e_{11}, e_{10}, -e_{13} )$ with $(e_7, e_{12}, e_{11})$ preserving signs and $(e_1, e_6, -e_{10}, e_{13})$ alternating signs.
$o_8 = ( e_2, e_4, e_8, e_6, e_{12}, -e_{14}, e_{10} )$
$o_9 = ( e_2, e_4, e_9, e_6, e_{13}, e_{15}, -e_{11}, -e_2, e_4, e_9, -e_6, e_{13}, -e_{15}, e_{11} )$ with $(e_4, e_9, e_{13})$ preserving signs and $(e_2, e_6, e_{15}, -e_{11})$ alternating signs.
$o_{10} = ( e_2, e_5, e_8, e_7, e_{13}, -e_{15}, e_{10} )$
$o_{11} = ( e_2, e_5, e_{12}, e_7, e_9, -e_{11}, e_{14}, -e_2, e_5, e_{12}, -e_7, e_9, e_{11}, -e_{14} )$ with $(e_5, e_{12}, e_9)$ preserving signs and $(e_2, e_7, -e_{11}, e_{14})$ alternating signs.
$o_{12} = ( e_3, e_4, e_8, e_7, e_{12}, -e_{15}, e_{11} )$
$o_{13} = ( e_3, e_4, e_9, e_7, e_{13}, -e_{14}, e_{10}, -e_3, e_4, e_9, -e_7, e_{13}, e_{14}, -e_{10} )$ with $(e_4, e_9, e_{13})$ preserving signs and $(e_3, e_7, -e_{14}, e_{10})$ alternating signs.
$o_{14} = ( e_3, e_6, e_8, e_5, e_{14}, -e_{13}, e_{11} )$
$o_{15} = ( e_3, e_6, e_{12}, e_5, e_{10}, -e_9, e_{15}, -e_3, e_6, e_{12}, -e_5, e_{10}, e_9, -e_{15} )$ with $(e_6, e_{12}, e_{10})$ preserving signs and $(e_3, e_5, -e_9, e_{15})$ alternating signs.

As you can see, 8 of these octonion-tuples are equal in sign to the first octonion-tuple, but 7 of them have flipped signs, causing the octonion-tuple to be traversed twice.
I'll denote the traverse-once octonion-tuples as 'positive parity' and the traverse-twice octonion-tuples as 'negative parity'.

octonion-tuple, parity, elements:
$\begin{matrix} o_1: & + & (e_1, & e_2, & e_4, & e_3, & e_6, & -e_7, & e_5) \\ o_2: & + & (e_1, & e_2, & e_8, & e_3, & e_{10}, & -e_{11}, & e_9) \\ o_3: & - & (e_1, & e_2, & e_{14}, & e_3, & e_{12}, & e_{13}, & -e_{15}) \\ o_4: & + & (e_1, & e_4, & e_8, & e_5, & e_{12}, & -e_{13}, & e_9) \\ o_5: & - & (e_1, & e_4, & e_{10}, & e_5, & e_{14}, & -e_{15}, & e_{11}) \\ o_6: & + & (e_1, & e_7, & e_8, & e_6, & e_{15}, & -e_{14}, & e_9) \\ o_7: & - & (e_1, & e_7, & e_{12}, & e_6, & e_{11}, & -e_{10}, & e_{13}) \\ o_8: & + & (e_2, & e_4, & e_8, & e_6, & e_{12}, & -e_{14}, & e_{10}) \\ o_9: & - & (e_2, & e_4, & e_9, & e_6, & e_{13}, & e_{15}, & -e_{11}) \\ o_{10}: & + & (e_2, & e_5, & e_8, & e_7, & e_{13}, & -e_{15}, & e_{10}) \\ o_{11}: & - & (e_2, & e_5, & e_{12}, & e_7, & e_9, & -e_{11}, & e_{14}) \\ o_{12}: & + & (e_3, & e_4, & e_8, & e_7, & e_{12}, & -e_{15}, & e_{11}) \\ o_{13}: & - & (e_3, & e_4, & e_9, & e_7, & e_{13}, & -e_{14}, & e_{10}) \\ o_{14}: & + & (e_3, & e_6, & e_8, & e_5, & e_{14}, & -e_{13}, & e_{11}) \\ o_{15}: & - & (e_3, & e_6, & e_{12}, & e_5, & e_{10}, & -e_9, & e_{15}) \\ \end{matrix}$

8 of these octonion-tuples preserve their signs correctly as you follow the multiplication around the octonion-tuple, just as the octonion-tuple does.
These are denoted as a positive parity.
The other 7 involve sign-flips on four of their elements and therefore must cycle through the list of quternion-tuples exactly twice.

For example, of the positive parity octonion-tuples, you can pick any element, the 2nd, and the 4rd to form an octonion multiplication table.
$o_1$ matches the octonions.
$o_2$ likewise creates the Mobius strip:
$\begin{matrix} e_1 & \times e_2 & & = e_3 & & & & \\ & e_2 & \times e_8 & & = e_{10} & & & \\ & & e_8 & \times e_3 & & = -e_{11} & \\ & & & e_3 & \times e_{10} & & & = e_9 \\ = e_1 & & & & e_{10} & \times -e_{11} & \\ & = e_2 & & & & -e_{11} & \times e_9 \\ \times e_1 & & = e_8 & & & & & e_9 \\ \end{matrix}$

However, for the remaining negative parity 7 octonion-tuples, to follow the multiplication quternion-tuples around the elements of the octonion-tuple, the octonion-tuple must be repeated twice.

Consider octonion-tuple $o_3 = (e_1,e_2,e_{14},e_3,e_{12},e_{13},e_{15})$:
Following the multiplication around the octonion-tuple we come up with the sequence:
$( e_1, e_2, e_{14}, e_3, e_{12}, e_{13}, -e_{15}, -e_1, e_2, e_{14}, -e_3, e_{12}, -e_{13}, e_{15} )$:
$\begin{matrix} e_1 & \times e_2 & & = e_3 & & & & & & & & & & \\ & e_2 & \times e_{14} & & = e_{12} & & & & & & & & & \\ & & e_{14} & \times e_3 & & = e_{13} & & & & & & & & \\ & & & e_3 & \times e_{12} & & = -e_{15} & & & & & & & \\ & & & & e_{12} & \times e_{13} & & = -e_1 & & & & & & \\ & & & & & e_{13} & \times -e_{15} & & = e_2 & & & & & \\ & & & & & & -e_{15} & \times -e_1 & & = e_{14} & & & & \\ & & & & & & & -e_1 & \times e_2 & & = -e_3 & & & \\ & & & & & & & & e_2 & \times e_{14} & & = e_{12} & & \\ & & & & & & & & & e_{14} & \times -e_3 & & = -e_{13} & \\ & & & & & & & & & & -e_3 & \times e_{12} & & = e_{15} \\ = e_1 & & & & & & & & & & & e_{12} & \times -e_{13} & \\ & = e_2 & & & & & & & & & & & -e_{13} & \times e_{15} \\ \times e_1 & & = e_{14} & & & & & & & & & & & e_{15} \\ \end{matrix}$

As you can see from following the octonion-tuple $o_3$ around twice, the elemets $e_1, e_3, e_{13}, e_{15}$ change signs twice while the elements of the quternion-tuple $(e_2, e_{14}, e_{12})$ do not.

In each of the negative parity octonion-tuples there is a unique quternion-tuple of elements that doesn't change signs as the octonion-tuple is covered twice. This is the second quternion-tuple of the negative parity octonion-tuple:
octonion-tuple, quternion-tuple that does not change signs, elements that do change signs
$\begin{matrix} o_3 & ( e_2, e_{14}, e_{12} ), & ( e_1, e_3, e_{13}, e_{15} ) \\ o_5 & ( e_4, e_{10}, e_{14} ), & ( e_1, e_5, e_{15}, e_{11} ) \\ o_7 & ( e_7, e_{12}, e_{11} ), & ( e_1, e_6, e_{10}, e_{13} ) \\ o_9 & ( e_4, e_9, e_{13} ), & ( e_2, e_6, e_{15}, e_{11} ) \\ o_{11} & ( e_5, e_{12}, e_7 ), & ( e_2, e_7, e_{11}, e_{14} ) \\ o_{13} & ( e_4, e_{9}, e_{13} ), & ( e_3, e_7, e_{10}, e_{14} ) \\ o_{15} & ( e_6, e_{12}, e_{10} ), & ( e_3, e_5, e_9, e_{15} ) \\ \end{matrix}$

For what it's worth:
Elements $e_1, e_3, e_8, e_{15}$ are not found among these quternion-tuples preserved from sign flips.
Element $e_4, e_8, e_{12}$ are not found among those that flip signs.
So $e_8$ alone isn't found at all among the negative parity octonion-tuples.
Also for what it's worth, choosing different quternion-tuple arrangements for the octonion-tuple will change the element signs, and which quternion-tuple does not change signs, but there will always be one quternion-tuple in the negative parity octonion-tuples that does not change signs.

Each element is shared by each octonion-tuple exactly 7 times.
element, octonion-tuples which the element is present:
$\begin{matrix} e_1 \in & o_1, & o_2, & o_4, & o_3, & o_6, & o_7, & o_5 \\ e_2 \in & o_1, & o_2, & o_8, & o_3, & o_{10}, & o_{11}, & o_9 \\ e_3 \in & o_1, & o_2, & o_{14}, & o_3, & o_{12}, & o_{13}, & o_{15} \\ e_4 \in & o_1, & o_4, & o_8, & o_5, & o_{12}, & o_{13}, & o_9 \\ e_5 \in & o_1, & o_4, & o_{10}, & o_5, & o_{14}, & o_{15}, & o_{11} \\ e_6 \in & o_1, & o_7, & o_8, & o_6, & o_{15}, & o_{14}, & o_9 \\ e_7 \in & o_1, & o_7, & o_{12}, & o_6, & o_{11}, & o_{10}, & o_{13} \\ e_8 \in & o_2, & o_4, & o_8, & o_6, & o_{12}, & o_{14}, & o_{10} \\ e_9 \in & o_2, & o_4, & o_9, & o_6, & o_{13}, & o_{15}, & o_{11} \\ e_{10} \in & o_2, & o_5, & o_8, & o_7, & o_{13}, & o_{15}, & o_{10} \\ e_{11} \in & o_2, & o_5, & o_{12}, & o_7, & o_9, & o_{11}, & o_{14} \\ e_{12} \in & o_3, & o_4, & o_8, & o_7, & o_{12}, & o_{11}, & o_{15} \\ e_{13} \in & o_3, & o_4, & o_9, & o_7, & o_{13}, & o_{14}, & o_{10} \\ e_{14} \in & o_3, & o_6, & o_8, & o_5, & o_{14}, & o_{13}, & o_{11} \\ e_{15} \in & o_3, & o_6, & o_{12}, & o_5, & o_{10}, & o_9, & o_{15} \\ \end{matrix}$

As you can see, the indexes of the octonions that possess basis elements is identical to the indexes of basis elements within octonions. From this we can define a dual between e's and o's: $\star e_i = o_i$.
There is a one-to-one correlation between which elements are contained by a octonion-tuple and which octonion-tuples are contained by an element, similar to the one-to-one relationship between elements and quternion-tuples found in the octonion multiplication table.

Each quternion-tuple of elements shares a quternion-tuple of octonion-tuples in common. Let's look at the octonion-tuples shared by each pair (the 3rd one follows suit):
$\begin{array}{c|cccccccccccccccc} & e_1 & e_2 & e_3 & e_4 & e_5 & e_6 & e_7 & e_8 & e_9 & e_{10} & e_{11} & e_{12} & e_{13} & e_{14} & e_{15} \\ \hline e_1 & & ( o_1, o_2, o_3 ) & ( o_1, o_2, o_3 ) & ( o_1, o_4, o_5 ) & ( o_1, o_4, o_5 ) & ( o_1, o_7, o_6 ) & ( o_1, o_7, o_6 ) & ( o_2, o_4, o_6 ) & ( o_2, o_4, o_6 ) & ( o_2, o_5, o_7 ) & ( o_2, o_5, o_7 ) & ( o_3, o_4, o_7 ) & ( o_3, o_4, o_7 ) & ( o_3, o_6, o_5 ) & ( o_3, o_6, o_5 ) \\ e_2 & ( o_1, o_2, o_3 ) & & ( o_1, o_2, o_3 ) & ( o_1, o_8, o_9 ) & ( o_1, o_{11}, o_{10} ) & ( o_1, o_8, o_9 ) & ( o_1, o_{11}, o_{10} ) & ( o_2, o_8, o_{10} ) & ( o_2, o_9, o_{11} ) & ( o_2, o_8, o_{10} ) & ( o_2, o_9, o_{11} ) & ( o_3, o_8, o_{11} ) & ( o_3, o_{10}, o_9 ) & ( o_3, o_8, o_{11} ) & ( o_3, o_{10}, o_9 ) \\ e_3 & ( o_1, o_2, o_3 ) & ( o_1, o_2, o_3 ) & & ( o_1, o_{13}, o_{12} ) & ( o_1, o_{14}, o_{15} ) & ( o_1, o_{14}, o_{15} ) & ( o_1, o_{13}, o_{12} ) & ( o_2, o_{14}, o_{12} ) & ( o_2, o_{15}, o_{13} ) & ( o_2, o_{15}, o_{13} ) & ( o_2, o_{14}, o_{12} ) & ( o_3, o_{15}, o_{12} ) & ( o_3, o_{13}, o_{14} ) & ( o_3, o_{13}, o_{14} ) & ( o_3, o_{15}, o_{12} ) \\ e_4 & ( o_1, o_4, o_5 ) & ( o_1, o_8, o_9 ) & ( o_1, o_{13}, o_{12} ) & & ( o_1, o_4, o_5 ) & ( o_1, o_8, o_9 ) & ( o_1, o_{13}, o_{12} ) & ( o_4, o_8, o_{12} ) & ( o_4, o_9, o_{13} ) & ( o_5, o_8, o_{13} ) & ( o_5, o_{12}, o_9 ) & ( o_4, o_8, o_{12} ) & ( o_4, o_9, o_{13} ) & ( o_5, o_8, o_{13} ) & ( o_5, o_{12}, o_9 ) \\ e_5 & ( o_1, o_4, o_5 ) & ( o_1, o_{11}, o_{10} ) & ( o_1, o_{14}, o_{15} ) & ( o_1, o_4, o_5 ) & & ( o_1, o_{14}, o_{15} ) & ( o_1, o_{11}, o_{10} ) & ( o_4, o_{10}, o_{14} ) & ( o_4, o_{11}, o_{15} ) & ( o_5, o_{10}, o_{15} ) & ( o_5, o_{14}, o_{11} ) & ( o_4, o_{11}, o_{15} ) & ( o_4, o_{10}, o_{14} ) & ( o_5, o_{14}, o_{11} ) & ( o_5, o_{10}, o_{15} ) \\ e_6 & ( o_1, o_7, o_6 ) & ( o_1, o_8, o_9 ) & ( o_1, o_{14}, o_{15} ) & ( o_1, o_8, o_9 ) & ( o_1, o_{14}, o_{15} ) & & ( o_1, o_7, o_6 ) & ( o_6, o_8, o_{14} ) & ( o_6, o_{15}, o_9 ) & ( o_7, o_8, o_{15} ) & ( o_7, o_9, o_{14} ) & ( o_7, o_8, o_{15} ) & ( o_7, o_9, o_{14} ) & ( o_6, o_8, o_{14} ) & ( o_6, o_{15}, o_9 ) \\ e_7 & ( o_1, o_7, o_6 ) & ( o_1, o_{11}, o_{10} ) & ( o_1, o_{13}, o_{12} ) & ( o_1, o_{13}, o_{12} ) & ( o_1, o_{11}, o_{10} ) & ( o_1, o_7, o_6 ) & & ( o_6, o_{12}, o_{10} ) & ( o_6, o_{11}, o_{13} ) & ( o_7, o_{13}, o_{10} ) & ( o_7, o_{12}, o_{11} ) & ( o_7, o_{12}, o_{11} ) & ( o_7, o_{13}, o_{10} ) & ( o_6, o_{11}, o_{13} ) & ( o_6, o_{12}, o_{10} ) \\ e_8 & ( o_2, o_4, o_6 ) & ( o_2, o_8, o_{10} ) & ( o_2, o_{14}, o_{12} ) & ( o_4, o_8, o_{12} ) & ( o_4, o_{10}, o_{14} ) & ( o_6, o_8, o_{14} ) & ( o_6, o_{12}, o_{10} ) & & ( o_2, o_4, o_6 ) & ( o_2, o_8, o_{10} ) & ( o_2, o_{14}, o_{12} ) & ( o_4, o_8, o_{12} ) & ( o_4, o_{10}, o_{14} ) & ( o_6, o_8, o_{14} ) & ( o_6, o_{12}, o_{10} ) \\ e_9 & ( o_2, o_4, o_6 ) & ( o_2, o_9, o_{11} ) & ( o_2, o_{15}, o_{13} ) & ( o_4, o_9, o_{13} ) & ( o_4, o_{11}, o_{15} ) & ( o_6, o_{15}, o_9 ) & ( o_6, o_{11}, o_{13} ) & ( o_2, o_4, o_6 ) & & ( o_2, o_{15}, o_{13} ) & ( o_2, o_9, o_{11} ) & ( o_4, o_{11}, o_{15} ) & ( o_4, o_9, o_{13} ) & ( o_6, o_{11}, o_{13} ) & ( o_6, o_{15}, o_9 ) \\ e_{10} & ( o_2, o_5, o_7 ) & ( o_2, o_8, o_{10} ) & ( o_2, o_{15}, o_{13} ) & ( o_5, o_8, o_{13} ) & ( o_5, o_{10}, o_{15} ) & ( o_7, o_8, o_{15} ) & ( o_7, o_{13}, o_{10} ) & ( o_2, o_8, o_{10} ) & ( o_2, o_{15}, o_{13} ) & & ( o_2, o_5, o_7 ) & ( o_7, o_8, o_{15} ) & ( o_7, o_{13}, o_{10} ) & ( o_5, o_8, o_{13} ) & ( o_5, o_{10}, o_{15} ) \\ e_{11} & ( o_2, o_5, o_7 ) & ( o_2, o_9, o_{11} ) & ( o_2, o_{14}, o_{12} ) & ( o_5, o_{12}, o_9 ) & ( o_5, o_{14}, o_{11} ) & ( o_7, o_9, o_{14} ) & ( o_7, o_{12}, o_{11} ) & ( o_2, o_{14}, o_{12} ) & ( o_2, o_9, o_{11} ) & ( o_2, o_5, o_7 ) & & ( o_7, o_{12}, o_{11} ) & ( o_7, o_9, o_{14} ) & ( o_5, o_{14}, o_{11} ) & ( o_5, o_{12}, o_9 ) \\ e_{12} & ( o_3, o_4, o_7 ) & ( o_3, o_8, o_{11} ) & ( o_3, o_{15}, o_{12} ) & ( o_4, o_8, o_{12} ) & ( o_4, o_{11}, o_{15} ) & ( o_7, o_8, o_{15} ) & ( o_7, o_{12}, o_{11} ) & ( o_4, o_8, o_{12} ) & ( o_4, o_{11}, o_{15} ) & ( o_7, o_8, o_{15} ) & ( o_7, o_{12}, o_{11} ) & & ( o_3, o_4, o_7 ) & ( o_3, o_8, o_{11} ) & ( o_3, o_{15}, o_{12} ) \\ e_{13} & ( o_3, o_4, o_7 ) & ( o_3, o_{10}, o_9 ) & ( o_3, o_{13}, o_{14} ) & ( o_4, o_9, o_{13} ) & ( o_4, o_{10}, o_{14} ) & ( o_7, o_9, o_{14} ) & ( o_7, o_{13}, o_{10} ) & ( o_4, o_{10}, o_{14} ) & ( o_4, o_9, o_{13} ) & ( o_7, o_{13}, o_{10} ) & ( o_7, o_9, o_{14} ) & ( o_3, o_4, o_7 ) & & ( o_3, o_{13}, o_{14} ) & ( o_3, o_{10}, o_9 ) \\ e_{14} & ( o_3, o_6, o_5 ) & ( o_3, o_8, o_{11} ) & ( o_3, o_{13}, o_{14} ) & ( o_5, o_8, o_{13} ) & ( o_5, o_{14}, o_{11} ) & ( o_6, o_8, o_{14} ) & ( o_6, o_{11}, o_{13} ) & ( o_6, o_8, o_{14} ) & ( o_6, o_{11}, o_{13} ) & ( o_5, o_8, o_{13} ) & ( o_5, o_{14}, o_{11} ) & ( o_3, o_8, o_{11} ) & ( o_3, o_{13}, o_{14} ) & & ( o_3, o_6, o_5 ) \\ e_{15} & ( o_3, o_6, o_5 ) & ( o_3, o_{10}, o_9 ) & ( o_3, o_{15}, o_{12} ) & ( o_5, o_{12}, o_9 ) & ( o_5, o_{10}, o_{15} ) & ( o_6, o_{15}, o_9 ) & ( o_6, o_{12}, o_{10} ) & ( o_6, o_{12}, o_{10} ) & ( o_6, o_{15}, o_9 ) & ( o_5, o_{10}, o_{15} ) & ( o_5, o_{12}, o_9 ) & ( o_3, o_{15}, o_{12} ) & ( o_3, o_{10}, o_9 ) & ( o_3, o_6, o_5 ) & \\ \end{array}$

You will notice that this table mostly matches with the element quternion-tuples, with some exceptions.
Some quternion-tuple indexes directly match while others are exchanged 1-1:
quternion-tuples of e #s:, quternion-tuples of octonion-tuple #s:
indexes that directly matching:
$\begin{matrix} ( e_1, e_2, e_3 ) & ( o_1, o_2, o_3 ) \\ ( e_1, e_4, e_5 ) & ( o_1, o_4, o_5 ) \\ ( e_1, e_7, e_6 ) & ( o_1, o_7, o_6 ) \\ ( e_2, e_8, e_{10} ) & ( o_2, o_8, o_{10} ) \\ ( e_2, e_9, e_{11} ) & ( o_2, o_9, o_{11} ) \\ ( e_3, e_{13}, e_{14} ) & ( o_3, o_{13}, o_{14} ) \\ ( e_3, e_{15}, e_{12} ) & ( o_3, o_{15}, o_{12} ) \\ ( e_4, e_8, e_{12} ) & ( o_4, o_8, o_{12} ) \\ ( e_4, e_9, e_{13} ) & ( o_4, o_9, o_{13} ) \\ ( e_5, e_{10}, e_{15} ) & ( o_5, o_{10}, o_{15} ) \\ ( e_5, e_{14}, e_{11} ) & ( o_5, o_{14}, o_{11} ) \\ ( e_6, e_8, e_{14} ) & ( o_6, o_8, o_{14} ) \\ ( e_6, e_{15}, e_9 ) & ( o_6, o_{15}, o_9 ) \\ ( e_7, e_{12}, e_{11} ) & ( o_7, o_{12}, o_{11} ) \\ ( e_7, e_{13}, e_{10} ) & ( o_7, o_{13}, o_{10} ) \\ \end{matrix}$
indexes that are paired:
$\begin{matrix} ( e_1, e_8, e_9 ) & ( o_2, o_4, o_6 ) \\ ( e_2, e_4, e_6 ) & ( o_1, o_8, o_9 ) \\ ( e_1, e_{11}, e_{10} ) & ( o_2, o_5, o_7 ) \\ ( e_2, e_5, e_7 ) & ( o_1, o_{11}, o_{10} ) \\ ( e_1, e_{13}, e_{12} ) & ( o_3, o_4, o_7 ) \\ ( e_3, e_4, e_7 ) & ( o_1, o_{13}, o_{12} ) \\ ( e_1, e_{14}, e_{15} ) & ( o_3, o_6, o_5 ) \\ ( e_3, e_6, e_5 ) & ( o_1, o_{14}, o_{15} ) \\ ( e_2, e_{14}, e_{12} ) & ( o_3, o_8, o_{11} ) \\ ( e_3, e_8, e_{11} ) & ( o_2, o_{14}, o_{12} ) \\ ( e_2, e_{15}, e_{13} ) & ( o_3, o_{10}, o_9 ) \\ ( e_3, e_{10}, e_9 ) & ( o_2, o_{15}, o_{13} ) \\ ( e_4, e_{10}, e_{14} ) & ( o_5, o_8, o_{13} ) \\ ( e_5, e_8, e_{13} ) & ( o_4, o_{10}, o_{14} ) \\ ( e_4, e_{11}, e_{15} ) & ( o_5, o_{12}, o_9 ) \\ ( e_5, e_{12}, e_9 ) & ( o_4, o_{11}, o_{15} ) \\ ( e_6, e_{11}, e_{13} ) & ( o_7, o_9, o_{14} ) \\ ( e_7, e_9, e_{14} ) & ( o_6, o_{11}, o_{13} ) \\ ( e_6, e_{12}, e_{10} ) & ( o_7, o_8, o_{15} ) \\ ( e_7, e_8, e_{15} ) & ( o_6, o_{12}, o_{10} ) \\ \end{matrix}$

It turns out if we collect all octonion-tuples that share a quaternion-tuple in common and follow the octonion-tuple multiplication loop we find something.
First let's look at all octonion-tuples that possess the first quaternion-tuple, $(e_1, e_2, e_3)$:
Then for the octonions that possess the next quaternion of $o_1$, which is $(e_2, e_4, e_6)$:
...etc...
$e_1, e_2, e_3 $$\in q_1 $$\in \left\{ \begin{matrix} o_1 = & ( e_1, & e_2, & e_4, & e_3, & e_6, & -e_7, & e_5 ) \\ o_2 = & ( e_1, & e_2, & e_8, & e_3, & e_{10}, & -e_{11}, & e_9 ) \\ o_3 = & ( e_1, & e_2, & e_{14}, & e_3, & e_{12}, & e_{13}, & -e_{15}, & -... ) \\ \end{matrix} \right.$
$e_2, e_4, e_6 $$\in q_4 $$\in \left\{ \begin{matrix} o_1 = & ( e_2, & e_4, & e_3, & e_6, & -e_7, & e_5, & e_1 ) \\ o_8 = & ( e_2, & e_4, & e_8, & e_6, & e_{12}, & -e_{14}, & e_{10} ) \\ o_9 = & ( e_2, & e_4, & e_9, & e_6, & e_{13}, & e_{15}, & -e_{11}, & -... ) \\ \end{matrix} \right.$
$e_4, e_3, -e_7 $$\in q_6 $$\in \left\{ \begin{matrix} o_1 = & ( e_4, & e_3, & e_6, & -e_7, & e_5, & e_1, & e_2 ) \\ o_{12} = & ( e_4, & e_3, & e_8, & -e_7, & e_{11}, & e_{15}, & e_{12} ) \\ o_{13} = & ( e_4, & e_3, & e_9, & -e_7, & -e_{10}, & e_{14}, & -e_{13}, & -... ) \\ \end{matrix} \right.$
$e_3, e_6, e_5 $$\in q_7 $$\in \left\{ \begin{matrix} o_1 = & ( e_3, & e_6, & -e_7, & e_5, & e_1, & e_2, & e_4 ) \\ o_{14} = & ( e_3, & e_6, & e_8, & e_5, & e_{14}, & -e_{13}, & e_{11} ) \\ o_{15} = & ( e_3, & e_6, & e_{12}, & e_5, & e_{10}, & -e_9, & e_{15}, & -... ) \\ \end{matrix} \right.$
$e_6, -e_7, e_1 $$\in q_3 $$\in \left\{ \begin{matrix} o_1 = & ( e_6, & -e_7, & e_5, & e_1, & e_2, & e_4, & e_3 ) \\ o_6 = & ( e_6, & -e_7, & e_8, & e_1, & -e_{15}, & -e_9, & e_{14} ) \\ o_7 = & ( e_6, & -e_7, & e_{12}, & e_1, & -e_{11}, & e_{13}, & -e_{10}, & -... ) \\ \end{matrix} \right.$
$-e_7, e_5, e_2 $$\in q_5 $$\in \left\{ \begin{matrix} o_1 = & ( -e_7, & e_5, & e_1, & e_2, & e_4, & e_3, & e_6 ) \\ o_{10} = & ( -e_7, & e_5, & e_8, & e_2, & e_{13}, & -e_{10}, & -e_{15} ) \\ o_{11} = & ( -e_7, & e_5, & e_{12}, & e_2, & e_9, & e_{14}, & e_{11}, & -... ) \\ \end{matrix} \right.$
$e_5, e_1, e_4 $$\in q_2 $$\in \left\{ \begin{matrix} o_1 = & ( e_5, & e_1, & e_2, & e_4, & e_3, & e_6, & -e_7 ) \\ o_4 = & ( e_5, & e_1, & e_8, & e_4, & e_9, & -e_{12}, & e_{13} ) \\ o_5 = & ( e_5, & e_1, & e_{10}, & e_4, & -e_{11}, & -e_{14}, & -e_{15}, & -... ) \\ \end{matrix} \right.$

Notice how every single element in the sedenions appears exactly once, except for the quaternion-tuple's elements which appear 3 times.
Notice how, for an octonion, of the 7 quaternions in the octonion, it shares each quaternion in common with exactly 2 other octonions, such that 2 x 7 = 14 remaining octonions among the 15 octonions in the sedenions.
As stated before: sometimes the basis element triplets and octonion index triplets match, sometimes they do not.
It's also interesting that the other two pairs of octonions are always sequential neighbors in our current indexing, i.e. $(o_2, o_3), (o_4, o_5), ...$.
It looks like the following triplets of elements and triplets of octonions match up: $(e_1, e_2, e_3), (e_1, e_4, e_5), (e_1, e_7, e_6)$.

Each pair of octonion-tuples shares a quternion-tuple of elements in common:
If you follow the same process used to find the 1st, 2nd, 4th spacing of the octonions, except use it to find spacings of equal shared objects between the 7 nested octonion elements of the 15 octonion-tuples in sedenions,
that all share equal neighbors, the you find the following spacings share 3 neighbors each:
spacing vector description
1st, 2nd, 3rd, 5th, 6th, 9th, 11th $\left[\begin{matrix} 1& 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \end{matrix}\right]$
1st, 2nd, 3rd, 8th, 10th, 13th, 14th $\left[\begin{matrix} 1& 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 \end{matrix}\right]$ rotation equivalent to order #1
1st, 2nd, 4th, 5th, 6th, 11th, 13th $\left[\begin{matrix} 1& 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \end{matrix}\right]$ shifted equivalent to order #2
1st, 2nd 4th 5th 8th 10th 15th $\left[\begin{matrix} 1& 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \end{matrix}\right]$ shifted equivalent to order #1
1st, 2nd 5th 7th 12th 13th 14th $\left[\begin{matrix} 1& 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \end{matrix}\right]$ shifted equivalent to order #1
1st, 2nd, 7th, 9th, 12th, 13th, 15th $\left[\begin{matrix} 1& 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 \end{matrix}\right]$ shifted equivalent to order #2
1st, 3rd, 4th, 5th, 10th, 12th, 15th $\left[\begin{matrix} 1& 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \end{matrix}\right]$ shifted equivalent to order #2
1st, 3rd, 4th, 7th, 9th, 14th, 15th $\left[\begin{matrix} 1& 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix}\right]$ shifted equivalent to order #1
1st, 3rd, 6th, 7th, 9th, 10th, 11th $\left[\begin{matrix} 1& 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \end{matrix}\right]$ shifted equivalent to order #2
1st, 3rd, 8th, 9th, 10th, 12th, 13th $\left[\begin{matrix} 1& 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \end{matrix}\right]$ shifted equivalent to order #1
1st, 4th, 5th, 7th, 8th, 9th, 14th $\left[\begin{matrix} 1& 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}\right]$ shifted equivalent to order #2
1st, 4th, 6th, 11th, 12th, 13th, 15th $\left[\begin{matrix} 1& 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \end{matrix}\right]$ shifted equivalent to order #1
1st, 6th, 7th, 8th, 10th, 11th, 14th $\left[\begin{matrix} 1& 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \end{matrix}\right]$ shifted equivalent to order #1
1st, 6th, 8th, 11th, 12th, 14th, 15th $\left[\begin{matrix} 1& 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \end{matrix}\right]$ shifted equivalent to order #2


There only one unique isomorphism of these spacings:
1st, 2nd, 3rd, 5th, 6th, 9th, 11th.

Using this spacing you can represent the sedenions as a unique list.
No attention is paid to signs, since the signs of the nested octonions occasionally flip.
When considering the embedded octonions, let's flip the 6th and 7th elements:

From the first octonion-tuple enumeration of $\begin{matrix} ( e_1 & e_2 & e_4 & e_3 & e_6 & -e_7 & e_5 ) \end{matrix}$:
$\begin{matrix} ( e_1 & e_2 & e_4 & \cdot & e_3 & e_6 & \cdot & \cdot & e_5 & \cdot & e_7 & \cdot & \cdot & \cdot & \cdot ) \\ ( \cdot & e_2 & e_4 & e_8 & \cdot & e_6 & e_{12} & \cdot & \cdot & e_{10} & \cdot & e_{14} & \cdot & \cdot & \cdot ) \\ fails \\ ... \\ ( \cdot & e_2 & e_4 & e_9 & \cdot & e_6 & e_{13} & \cdot & \cdot & e_{11} & \cdot & e_{15} & \cdot & \cdot & \cdot ) \\ ( \cdot & \cdot & e_4 & e_9 & \cdot & e_6 & e_{13} & \cdot & \cdot & e_{11} & \cdot & e_{15} & \cdot & \cdot & \cdot ) \\ fails \\ ... \\ \end{matrix}$
From the second octonion-tuple enumeration of $\begin{matrix} ( e_1 & e_2 & e_8 & e_3 & e_{10} & -e_{11} & e_9 ) \end{matrix}$:
$\begin{matrix} ( e_1 & e_2 & e_8 & \cdot & e_3 & e_{10} & \cdot & \cdot & e_9 & \cdot & e_{11} & \cdot & \cdot & \cdot & \cdot ) \\ ( \cdot & e_2 & e_8 & e_8 & \cdot & e_7 & e_{13} & \cdot & \cdot & e_{10} & \cdot & e_{15} & \cdot & \cdot & \cdot ) & fails \\ ( \cdot & \cdot & e_5 & e_8 & ...\\ (e_1 & e_2 & e_5 & e_8 & e_3 & e_7 & e_{13} & e_{11} & e_4 & e_{10} & e_6 & e_{15} & e_{14} & e_{12} & e_9 ) works \end{matrix}$.


...such that we can come up with the sub-octonion table:
$\begin{matrix} s = & (e_1 & e_2 & e_5 & e_8 & e_3 & e_7 & e_{13} & e_{11} & e_4 & e_{10} & e_6 & e_{15} & e_{14} & e_{12} & e_9) \\ \hline o_1 = & e_1 & e_2 & e_5 & \cdot & e_3 & e_7 & \cdot & \cdot & e_4 & \cdot & e_6 & \cdot & \cdot & \cdot & \cdot \\ o_{10}=& \cdot & e_2 & e_5 & e_8 & \cdot & e_7 & e_{13} & \cdot & \cdot & e_{10} & \cdot & e_{15} & \cdot & \cdot & \cdot \\ o_{14}=& \cdot & \cdot & e_5 & e_8 & e_3 & \cdot & e_{13} & e_{11} & \cdot & \cdot & e_6 & \cdot & e_{14} & \cdot & \cdot \\ o_{12}=& \cdot & \cdot & \cdot & e_8 & e_3 & e_7 & \cdot & e_{11} & e_4 & \cdot & \cdot & e_{15} & \cdot & e_{12} & \cdot \\ o_{13}=& \cdot & \cdot & \cdot & \cdot & e_3 & e_7 & e_{13} & \cdot & e_4 & e_{10} & \cdot & \cdot & e_{14} & \cdot & e_9 \\ o_7 = & e_1 & \cdot & \cdot & \cdot & \cdot & e_7 & e_{13} & e_{11} & \cdot & e_{10} & e_6 & \cdot & \cdot & e_{12} & \cdot \\ o_9 = & \cdot & e_2 & \cdot & \cdot & \cdot & \cdot & e_{13} & e_{11} & e_4 & \cdot & e_6 & e_{15} & \cdot & \cdot & e_9 \\ o_5 = & e_1 & \cdot & e_5 & \cdot & \cdot & \cdot & \cdot & e_{11} & e_4 & e_{10} & \cdot & e_{15} & e_{14} & \cdot & \cdot \\ o_8 = & \cdot & e_2 & \cdot & e_8 & \cdot & \cdot & \cdot & \cdot & e_4 & e_{10} & e_6 & \cdot & e_{14} & e_{12} & \cdot \\ o_{15}=& \cdot & \cdot & e_5 & \cdot & e_3 & \cdot & \cdot & \cdot & \cdot & e_{10} & e_6 & e_{15} & \cdot & e_{12} & e_9 \\ o_6 = & e_1 & \cdot & \cdot & e_8 & \cdot & e_7 & \cdot & \cdot & \cdot & \cdot & e_6 & e_{15} & e_{14} & \cdot & e_9 \\ o_3 = & e_1 & e_2 & \cdot & \cdot & e_3 & \cdot & e_{13} & \cdot & \cdot & \cdot & \cdot & e_{15} & e_{14} & e_{12} & \cdot \\ o_{11}=& \cdot & e_2 & e_5 & \cdot & \cdot & e_7 & \cdot & e_{11} & \cdot & \cdot & \cdot & \cdot & e_{14} & e_{12} & e_9 \\ o_4 = & e_1 & \cdot & e_5 & e_8 & \cdot & \cdot & e_{13} & \cdot & e_4 & \cdot & \cdot & \cdot & \cdot & e_{12} & e_9 \\ o_2 = & e_1 & e_2 & \cdot & e_8 & e_3 & \cdot & \cdot & e_{11} & \cdot & e_{10} & \cdot & \cdot & \cdot & \cdot & e_9 \\ \end{matrix}$

So now we have a mobius-strip of mobius-strips. What is that going to look like...

overall

We see a dual between the elements of the Cayley-Dickson construction:
construction
elements
quaternions
octonions
sedenions
quaternions 3 1
octonions 7 7 1
sedenions 15 35 15 1


This turns out to be "Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2" (Wikipedia) (Wolfram MathWorld)