This is from papers about 1940 by Gordon Pall, one with B. W. Jones. I'm looking for statements about things being primitive, especially odd/even.

You did not mention this, so, in case this will make dimension 3 neater, all rational orthogonal matrices come from integers $a^2 + b^2 + c^2 + d^2 = n$ and the standard matrix describing quaternions, $$ \frac{1}{n} \; \left( \begin{array}{ccc} a^2 + b^2 - c^2 - d^2 & 2(-ad+bc) & 2(ac+bd) \\ 2(ad+bc) & a^2 - b^2 + c^2 - d^2 & 2(-ab+cd) \\ 2(-ac+bd) & 2(ab+cd) & a^2 - b^2 - c^2 + d^2 \\ \end{array} \right) $$

This is from papers about 1940 by Gordon Pall, one with B. W. Jones. I'm looking for statements about things being primitive, especially odd/even. Found it, also in "Rational Automorphs," in order to ge6t the gcd of the nine integer elements and $n$ to be $1,$ we have $n$ odd. This is Theorem 1 on page 754

You did not mention this, so, in case this will make dimension 3 neater, all rational orthogonal matrices come from integers $a^2 + b^2 + c^2 + d^2 = n$ and the standard matrix describing quaternions, $$ \frac{1}{n} \; \left( \begin{array}{ccc} a^2 + b^2 - c^2 - d^2 & 2(-ad+bc) & 2(ac+bd) \\ 2(ad+bc) & a^2 - b^2 + c^2 - d^2 & 2(-ab+cd) \\ 2(-ac+bd) & 2(ab+cd) & a^2 - b^2 - c^2 + d^2 \\ \end{array} \right) $$