What's the distribution of $xy+xz+yz$ where $x,y,z $ are independent standard normal? We know the product of two independent Normal random variables has a normal product distribution, or Variance Gamma distribution if they are correlated.
But, what if there are three Normal random variables? 
So, here is the question: Suppose $x,y,z$ are three independent normal random variables ($x, y, z\sim N(0,1)$), what's the distribution of $xy+xz+yz$?

Not a full answer, but I wanted to be able to provide an image.  I did $10^8$ simulations of this distribution and plotted a density histogram.  Just looking at the shape suggests the PDF is not simple; the asymmetry was surprising to me at first but it makes sense upon some reflection.

Frankly, I was surprised at how quickly Mathematica performed the simulation.  Its implementation of standard normal variates must undoubtedly be extremely efficient.
