Find the smallest positive integer $m$ such that there exist six distinct positive integers $\{a,b,c,d,e,f\}$ satisfying $$a^2+b^2+c^2 = d^2+e^2+f^2 = m^2\\ ad + be = cf$$ or show that this cannot be satisfied using six distinct positive integers, for any $m$.
For example, with $\{a=2,b=11,c=10,d=14,e=2,f=5\}$ we have $$ 2^2+11^2+10^2 = 14^2+2^2+5^2 = 225 = 15^2 \\ 2\cdot 14 + 11\cdot 2 = 10\cdot 5 $$ so this set would work, except that here $e = a = 2$ so they are not all distinct.
In the order you want, $$ \left( \begin{array}{ccc} 13 & 34 & 14 \\ 26 & 2 & 29 \end{array} \right) $$
Your $m=39$
The first rational orthogonal matrix with all entry absolute values distinct has your $m=57$
$$ \frac{1}{57} \; \left( \begin{array}{ccc} 17 & 16 & 52 \\ 32 & -47 & 4 \\ 44 & 28 & -23 \\ \end{array} \right) $$