Cayley's hyperdeterminant, the principal minors of a symmetric matrix and the entropy region of 4 Gaussian random variables

It has recently been shown that there is a connection between Cayley's hypdeterminant and the principal minors of a symmetric matrix. With an eye towards characterizing the entropy region of jointly Gaussian random variables, we obtain three new results on the relationship between Gaussian random variables and the hyperdeterminant. The first is a new (determinant) formula for the 2times2times2 hyperdeterminant. The second is a new (transparent) proof of the fact that the principal minors of an ntimesn symmetric matrix satisfy the 2 times 2 times .... times 2 (n times) hyperdeterminant relations. The third is a minimal set of 5 equations that 15 real numbers must satisfy to be the principal minors of a 4 times 4 symmetric matrix.