This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual understanding and related references if they exist.
Let $R$ be a commutative ring. For two vectors $v=(a,b,c,d), w=(A,B,C,D)\in R^4$, we define $v\star w:= (aA,aB+bA,bB, cC,cD+dC,dD)\in R^6$. Given any 3 vectors $v_1,v_2,v_3\in R^4$, we can form a $6\times 6$ matrix $M$ whose rows are $v_i\star v_j$, $1\leq i,j\leq 3$. Then: $$\det(M)=0$$
It is not clear to me how to explain this. The kernel of $M$ is a column of degree $6$ polynomials, so the relations are quite complicated.
Question: Is there a way to conceptually explain the vanishing of $\det(M)$? Have you seen similar identities?