  The quadruple $(1\,484\,801, 1\,203\,120, 1\,169\,407, 1\,157\,520)$ already
known is essentially the only non-trivial solution of the Diophantine equation
$x^4 + 2 y^4 = z^4 + 4 w^4$ for $|x|$, $|y|$, $|z|$, and $|w|$ up to one
hundred million. We describe the algorithm we used in order to establish this
result, thereby explaining a number of improvements to our original approach.
