element of ${{\mathbb Z}}[y_{ij} \mid i \geq j]$. The ideal of $F_{2,6}$ contains another type of equation: the [*off-diagonal minor*]{} $$M:=\det(y[\{4,5,6\},\{1,2,3\}]) \in {{\mathbb Z}}[y_{ij} \mid i \geq j]$$ the determinant of the square submatrix of $y$ sitting in the lower left corner of $y$. If $f$ is any polynomial in ${{\mathbb R}}[y_{ij} \mid i \geq j]$ that vanishes on $F_{2,n}$ and if we regard $f$ as an element of ${{\mathbb R}}[y_{ij} \mid i>j][y_{11},\ldots,y_{nn}]$, then each of the coefficients of the monomials in the diagonal variables $y_{ii}$ is a polynomial in the off-diagonal variables that vanishes on $F_{2,n}$, as well. Therefore the following theorem settles the conjecture of Drton, Sturmfels, and Sullivant, that pentads and off-diagonal minors generate the ideal of $F_{2,n}$ for all $n$; see [@Drton07 Conjecture 26].

\[thm:Main\] For any field $K$ and any natural number $n \geq 6$ the kernel $I_n(K)$ of the homomorphism $K[y_{ij}\mid 1 \leq j < i \leq n] \to
K[s_1,\ldots,s_n,t_1,\ldots,t_n]$ determined by $y_{ij} \mapsto s_is_j +
t_it_j$ is generated, as an ideal, by the orbits of $P$ and $M$ under the symmetric group ${\operatorname{Sym}}(n)$.

In [@Drton08] it is proved that $F_{2,n}$ equals the set of all positive definite matrices with the property that every 