An example of a non-associative Moufang loop of point classes on a cubic surface

Let \(V\) be a cubic surface defined by the equation \(T_0^3+T_1^3+T_2^3+\theta T_3^3=0\) over a quadratic extension of 3-adic numbers \(k=\mathbb{Q}_3(\theta)\), where \(\theta^3=1\). We show that a relation on a set of geometric k-points on \(V\) modulo \((1-\theta)^3\) (in a ring of integers of \(k\)) defines an admissible relation and a commutative Moufang loop associated with classes of this admissible equivalence is non-associative. This answers a problem that was formulated by Yu. I. Manin more than 50 years ago about existence of a cubic surface with a non-associative Moufang loop of point classes.