Calculation of certain expectation

Question

Let $X=(x_1,x_2,x_3,x_4)^t \sim MultiNormal(0,\Sigma)$ where $\Sigma=(\sigma_{i,j})$ is a non-degenerate covariance matrix, then is there any simple way of calculating $Cov(X_1X_2,X_3X_4)$ without doing integration?(or simplified trick of doing this integration)

Answer 1

I'm afraid not. The distribution of the product of two normals is not pretty, let alone their covariance