In the question, $G\in R^{n\times n}$ is the symmetric positive semi-definite (SPSD) matrix, $\det(\cdot)$ is the determinant of the matrix, $G_I$ is a principal submatrix of the SPSD matrix $G$, and $|I|=k$ means that the size of the principal submatrix is $k$-by-$k$.
For $\sum_{|I|=k+1}\det(G_I)$, it can be represented by the eigenvalues of $G$, as shown below although I don't know how Cauchy–Binet Theorem is applied.
My question is for $\sum_{|I|=k}\det(G_I)^2$.