Background: The equation

$$a^4+b^4+c^4=2d^4$$

has infinitely many positive integral solutions if we take $c=a+b$ and $a^2+ab+b^2=d^2$ further assuming that $GCD(a,b,c)=1$. This has been proved by Elkies using elliptic curves.

Main problem: Find some positive integral solutions to the equation

$$a^4+b^4+c^4=2d^4$$

with $a\lt b\lt c\ne a+b$ and $GCD(a,b,c)=1$.