Let $V$ be a general smooth projective cubic hypersurface. Doing literally as in case of cubic curves we define a relation on $V\times V\times V$: $(x,y,z)$ satisfy it iff $x+y+z$ is an intersection of $V$ with a line. Contrary to the one-dimensional case this relation is not a graph of a binary operation ($x^2$ is not defined or two points may lie on the line contained in $V$). From now on let us consider only pairs $(x,y)$ for which $z$ is uniquely defined (that is $(x,y,z)$ are on a line and there is one such $z$, let's denote it by $x\circ z$). Let's chose some $u$ (that will be the 'unit') and define the product as usual $xy=u\circ (x\circ y)$. This partially defined product may be non-associative in the following sense: there are $x,y,z$ such that $$x(yz)\neq(xy)z$$ where $yz$,$xy$, $x(yz)$ and $(xy)z$ are uniquely defined.

Are there any examples of such non-associative triples?

Let $V$ be a general smooth projective cubic hypersurface. Doing literally as in case of cubic curves we define a relation on $V\times V\times V$: $(x,y,z)$ satisfy it iff $x+y+z$ is an intersection of $V$ with a line. Contrary to the one-dimensional case this relation is not a graph of a binary operation ($x^2$ is not defined or two points may lie on the line contained in $V$). From now on let us consider only pairs $(x,y)$ for which $z$ is uniquely defined (that is $(x,y,z)$ are on a line and there is one such $z$, let's denote it by $x\circ z$). Let's chose some $u$ (that will be the 'unit') and define the product as usual $xy=u\circ (x\circ y)$. This partially defined product however may be non-associative in the following sense: there are $x,y,z$ such that $$x(yz)\neq(xy)z$$ where $yz$,$xy$, $x(yz)$ and $(xy)z$ are uniquely defined.

Are there any examples of such non-associative triples?