As you appear to be unaware of the fact, there are infinitely, infinitely many cubic quadruples
the smallest of which is 6^3 = 5^3 + 4^3 + 3^3 for the positive series and 9^3 = 8^3 + 6^3 - 1^3
for the negative series.
There are infinitely many quartic quadruples.
An 'interesting' trio of quadruples is:
[1] 159^4 = 158^4 + 59^4 + 1952^2
[2] 159^4 = 134^4 + 133^4 + 1952^2
[3] 158^4 = 134^4 + 133^4 - 59^4
If a Fermat triple existed it would mean that a trio of quadruples, 2 positive
and 1 negative and of the same power would also exist but the structure
of their interrelationship would be marginally different.
The 3 quadruples reduce to 3 Pythagorean quadruples.