This page lists some examples of Adem relations, sorted by degree. The general formula is that when , with the understanding that if or , and if (so in practice ranges from to ).
I learnt the following "derivation rule" from Eric Peterson: all the Adem relations follow from and asserting that the map is a derivation.
"i don't know who this version is due to. i learned it from sean tilson who learned it from bob bruner to whom many mysterious things seem obvious" — Eric Peterson
Edited to add: Bob Bruner attributes it to Kristensen's On Secondary Cohomology Operations
A useful interactive calculator can be found at Alex Kruckman's webpage.