Polynomial relations among principal minors of a 4x4-matrix

The image of the principal minor map for n x n-matrices is shown to be closed. In the 19th century, Nansen and Muir studied the implicitization problem of finding all relations among principal minors when n=4. We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in ${\text{PP}}^{15}$. The latter is the main component in the singular locus of the 2 x 2 x 2 x 2-hyperdeterminant.