On $n$-derivations and Relations between Elements $r^{n}-r$ for Some $n$

We find complete sets of generating relations between the elements $[r]=r^n-r$ for $n=2^l$ and for $n=3$. One of these relations is the $n$-derivation property $[rs]=r^{n}[s]+s[r]$, $r,s \in R$.