I am not sure whether "Multivariate Fourth Moment" is the name for it. But basically, I have an N-dimensional vector x, which is Gaussian with mean 0 and covariance matrix R. I also have two N by N matrix A and B. What I what to do is to compute the expectation E[(x(transpose)Ax)(x(transpose)Bx)]. I have seen a derivation in a paper that express this in terms of the covariance R, but I don't understand the critical step of expressing the "fourth moment" in terms of covariance (i.e. the step from line 2 to 3). I would appreciate if someone could explain how this is done. Also, is this result true in general or just because x is Gaussian? Thanks.