Spin-1 chain decimation recursion ($W⟼W^2$)

For the spin inversion symmetry parameterization $W=c\left(\begin{array}{ccc}1& x& y\\ x& z& x\\ y& x& 1\end{array}\right)$ with positive entries, spin decimation gives the same form with $x^{\prime }=x(1+y+z)/(1+x^2+y^2)$, $y^{\prime }=(x^2+2y)/(1+x^2+y^2)$ and $z^{\prime }=(z^2+2x^2)/(1+x^2+y^2)$. Multiply $W$ by itself and divide by its $(+,+)$ entry to prove these ratios. The leftover scalar is an additive free-energy coupling; dropping it leaves normalized expectations unchanged but loses the full free energy.