A universal non-associative Moufang loop connected to a cubic surface

Let $V$ be the cubic surface defined by the equation $T_0^3+T_1^3+T_2^3+\theta T_3^3=0$ over the quadratic extension $k=\mathbb {Q}_3(\theta )$ of the 3-adic rationals, where $\theta ^3=1$. We show that a relation on $V(k)$ modulo $(1-\theta )^3$ (in the