<s>[INST] $a^2+b^2+c^2 = d^2+e^2+f^2 = m^2, ad + be = cf$ with $\{a,b,c,d,e,f\}$ all distinct positive integers 
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.
 [/INST] 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 $m=57$
$$
\frac{1}{57} \;
\left(
\begin{array}{ccc}
17 & 16 & 52 \\
32 & -47 & 4 \\
44 & 28 & -23 \\
\end{array}
\right)
$$
</s>