I think a sequence of the following $9$ permutations is no one-product sequence: $12$, $23$, $34$, $45$, $13$, $24$, $35$, $14$, $25$

You do have that in general the norm of a vector is invariant under multiplication with a Hadamard matrix that is downscaled to a unitary matrix. So also for eigenvectors, giving that eigenvalues have norm $m$.

Actually I don't think the group generated by these matrices will be a proper subgroup of $SO(8)$, because we have to use all 7 versions of integral octonions.

I think $SO(8)$ has rotations between integral unit octonions as proper subgroup strictly containing $W_8$.

You're right @AndreasBlass . It's called in general linear position, where usually the basis is $\{y_i - y_j\}_{j \neq i}$

I will probably update the question after having read myself into the topic a little

The drawing means projection on the 2D plane with the smallest number of crossings. I didn't mean $\Bbb CP^2$. The result appears to apply to both $\Bbb CP^1$ and $\Bbb RP^2$

Note: it may be that I mean $\Bbb{CP}^1$ instead of $\Bbb{RP}^2$

Somewhat related: For $p=k^2+k+1$ you can multiply with $k$ as involution of order $3$ not just of $x$ but of any element in the field of fractions.

What about gaps between prime powers? Then the case $3$ is covered.

I think it's best to demonstrate this difficulty by writing a computer program that tries to add such orthogonal rows randomly for a given dimension, and record the statistics at which d no new row can be found anymore. The fact that CHM's are hard to find is known, so that's not an issue.