Let $G$ be a nilpotent group of class 3. Then for every $x ,y ,z$ in $G$, $[x,y,z][y,z,x][z,x,y]=1$.
As $G$ is a nilpotent group of class 3, $[G,G,G,G]=1$ and $G^{'}$ is abelian. I want to use Hall-Witt alliance but I don't know how to choose the elements.
Hall-Witt alliance : $[x,y^{-1},z]^{y}[y,z^{-1},x]^{z}[z,x^{-1},y]^{x}=1$
We have $[x,y,z,y^{-1}] =1$. So $[x,y,z]=[x,y,z]^{y^{-1}}$. I don't know how to continue.