Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the i-th row and j-th column of $M$.

The Desnanot-Jacobi Identity states $$\det(M)\det(M^{1,n}_{1,n})=\det(M^1_1)\det(M^n_n)-\det(M^n_1)\det(M^1_n).$$

If you view $M\in A\otimes B$ where $A$ and $B$ are $n$ dimensional vectors spaces, then $\det(M)\in \Lambda^nA\otimes\Lambda^nB\subset S^n(A\otimes B)$ and the determinant of a $k\times k$ minor is an element of $\Lambda^kA\otimes\Lambda^kB\subset S^k(A\otimes B)$.

With this interpretation of matrices, determinants, and determinants of minors of matrices, I would like to have a geometric interpretation of the Desnanot-Jacobi Identity.

Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the $i$-th row and $j$-th column of $M$.

The Desnanot–Jacobi Identity states $$\det(M)\det(M^{1,n}_{1,n})=\det(M^1_1)\det(M^n_n)-\det(M^n_1)\det(M^1_n).$$

If you view $M\in A\otimes B$ where $A$ and $B$ are $n$-dimensional vector spaces, then $\det(M)\in \Lambda^nA\otimes\Lambda^nB\subset S^n(A\otimes B)$ and the determinant of a $k\times k$ minor is an element of $\Lambda^kA\otimes\Lambda^kB\subset S^k(A\otimes B)$.

With this interpretation of matrices, determinants, and determinants of minors of matrices, I would like to have a geometric interpretation of the Desnanot–Jacobi Identity.