The Isserlis theorem states that if [x,y,z,t] is a zero mean multivariate normal random vector, then :

E[xyzt] = E[xy]E[zt] + E[xz]E[yt] + E[xt]E[yz]

Now, in the case x,y,z,t are not necessarily Gaussian random variables but the variable a =xyzt is a Gaussian variable with a certain mean and variance, does the above relation still holds or ,if not, is there any other way of calculating the mean value of a, i.e. E[a] = E[xyzt], knowing the first and second order moments of x, y ,z and t?

Thank you for your help