Higher-order composition of a symmetric splitting

If $S_h$ is a symmetric second-order splitting, then $S_{\gamma h}{S}_{\delta h}{S}_{\gamma h}$ is fourth order for $\gamma =(2-2^{1/3}{)}^{-1}$ and $\delta =-2^{1/3}(2-2^{1/3}{)}^{-1}$. The identities $2\gamma +\delta =1$ and $2{\gamma }^3+{\delta }^3=0$ preserve the total step and cancel the leading odd defect.