X,Y and Z operators of some algebra

[zetafunction]

let be X,Y and Z operators of some algebra so

$$[X_i , Y_j]= \epsilon _ijkX_k$$ where i,j and k range over X, Y and Z

then i define the change of coordinates

$$X= rcos(u)$$

$$Y= rsin(u)$$

$$Z=Z$$

again r, u and Z are new operators, the problem is , how can i find for example what is the value of $$[r, u]$$ or $$[r,z]$$ or $$[u,z]$$ ??

 

[HallsofIvy]

If X, Y, and Z are in "some algebra", then what do "r cos(u)" and "r sin(u)" mean?