Computing Poisson brackets

Question

Suppose I have Hamiltonian :H = $\frac{1}{2}(S_1^2+S_2^2+\beta S_3^2)+R_1$, and two first integrals: $f_1 = R_1^2+R_2^2+R_3^2$,$f_2 = R_1S_1+R_2S_2+R_3S_3$. And also i know how to compute Poisson bracket: first i write down $w^{ij} = $ {$x^i,x^j$},$(x_1,x_2,x_3,x_4,x_5,x_6)=(S_1,S_2,S_3,R_1,R_2,R_3)$ and {$S_i,S_j$}=$\epsilon_{ijk}S_k$, {$R_i,S_j$}=$\epsilon_{ijk}R_k$, {$R_i,R_j$}=0 and to compute {f,g} = $\sum_{ij} w^{ij} \dfrac{f}{x^i} \dfrac{g}{x^j}$.So my problem is when i compute {$f_1,f_2$} i get {$f_1,f_2$} = $2R_1R_2R_3$, but i know that this functions are Kasimir functions , so there Poisson bracket must be zero.