Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ
We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= – e_i,$ extend by linearity. Then we get a norm, or squared Euclidean length, as
$Nx = x \bar{x} = \bar{x} x,$ and the $N(xy) = Nx \; Ny.$ We get the real part $\mathcal Rx = \frac{1}{2} (x + \bar{x}). $ We will call an octonion $o$ pure if its real part is $0,$ that is $\bar{o} = -o.$ We also retrieve the ordinary dot product with $(x,y)= \mathcal R(x\bar{y}).$ Among a million other facts, multiplication on either the left or the right by a fixed octonion preserves angles, and multiplication by any of the $e_i$ rotates everything $90^\circ.$
The main law we need is the "alternative algebra" law,
$$ (xy)x = x(yx). $$ I like the quick summary of the laws in
Robert Wilson 2008 SEMINAR PDF
LEMMA: $$ (\bar{t} x) t = \bar{t} (x t) $$
Proof: Write $t = r + o $ with $r$ real and $o$ pure. The $\bar{t} = r – o.$ Calculation gives
$$ (\bar{t} x) t = r^2 x + r x o – r o x – (ox)o, $$
while
$$ \bar{t} (x t) = r^2 x + r x o – r o x – o(xo). $$
Since $ (ox)o = o (xo) $ the lemma follows.
To the question of interest, which was done for the quaternions as the matrix in formula (10) on page 755 of Pall Automorphs (1940) and Theorem 3 on page 176 of Jones and Pall (1939), both available as pdfs at
TERNARY
Let
$$ o = o_1 e_1 + o_2 e_2 + o_3 e_3 + o_4 e_4 + o_5 e_5 + o_6 e_6 + o_7 e_7 $$
be a "proper" pure quaternion, that is $\gcd(o_1,o_2,o_3,o_4,o_5,o_6,o_7) = 1,$ with square norm
$$ No = m^2 = o_1^2 + o_2^2 + o_3^2 + o_4^2 + o_5^2 + o_6^2 + o_7^2 $$ for integer $m.$
QUESTION: is it necessarily the case that we can find some integral octonion $t,$ with $Nt = m,$ such that
$$ o = \pm \bar{t} (e_j t) $$ for some $j =1,2,3,4,5,6,7?$
EDDITTT: I got an email from a kind fellow who has a Maple package for playing with octonions, who says that with $o = e_4 + e_5 + e_6 + e_7$ of norm 4 has no expression as $ o = \bar{t} (e_1 t). $ I have asked him if checking all possible $\pm e_j$ makes any difference. It is a big help to be able to experiment, I don't have any computer programs myself this time. I will put in his name if he says that is alright. Oh, his Maple program may be using a slightly different multiplication table from the one I picked, i think there are 240 or 480 possible versions.
Note that the proof by Jones and Pall uses associativity and, essentially, unique factorization (Theorem 2) for a proof by induction on the number of prime factors in out $m.$ That proof cannot work in quite such a pleasant manner. That is, their Theorem 2 is likely false, but perhaps this (Theorem 3) can be recovered as a very special case. Note that I can live with $m$ being odd, Jones and Pall needed to do that. For seven squares I do not think it should be necessary.
Oh, automorphs. It is fairly likely that every rational automorph of the sum of seven squares can be written with the seven rows being $\bar{t} (e_i t),$ and then divide by $m.$ I'm just saying.
EDIT, JUNE 7: note that the question to which I was trying to apply octonions has an affirmative answer due to J. S. Hsia, see Filling in a rational orthogonal matrix given one row while the octonion approach to seven variables was just not very effective. That is, it works for some 7-tuples but the percentage of failures goes up as the coefficients increase in size.
Best Answer
I'm not sure whether you were exploring a particular application of the octonions, but I want to point out that understanding the spaces where $o=\pm \bar{t} ( e_j t )$ exists are interesting. Papers like http://arxiv.org/abs/1002.1497 or http://arxiv.org/abs/0911.2255 use such constructs, and you're wondering about the further restriction of $t$ and $o$ being a 'proper octonion'. Thanks for posting your question, and some investigation. I wonder what actually are the spaces where a $t$ exists for some $e_j$ such that $o=\pm \bar{t} ( e_j t )$ for a given $o$? Quick question: With "integral octonion" did you mean any octonion that is made from integral coefficients? Or did you mean the (arguably) more customary use where "integral octonions" refers to a (reschaled) version of the E$_8$ lattice? Thanks, Jens