This is from [papers about 1940 by Gordon Pall, one with B. W. Jones][1]. 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,
$$
\frac{1}{n} \;
\left(
\begin{array}{ccc}
a^2 + b^2 - c^2 - d^2 & 2(-ad+bc) & 2(ac+bd)  \\
2(ad+bc) & a^2 - b^2 + c^2 - d^2 & 2(-ab+cd)  \\
2(-ac+bd) &  2(ab+cd) & a^2 - b^2 - c^2 + d^2  \\
\end{array}
\right)
$$


  [1]: http://zakuski.math.utsa.edu/~kap/