Assuming two variables $X1$ ~ $N(0,1)$, $X2$ ~ $N(0,1)$
with $Cov(X1,X2) = a$.
Is it possible to derive analytically what the covariance between $X1^2$ and $X2^2$ would be? Empirically (I tried this with large simulations), it appears that
$Cov(X1^2,X2^2) = 2*(Cov(X1,X2))^2$
I think this may however be just a coincidence because variables X1 and X2 have mean 0, and would in general love to derive this analytically.