I'm trying to figure out second moment of the following quantity

$$y = \frac{<x_1, x_2>}{\|x_1\|\|x_2\|}$$

Where $x_1$, $x_2$ are sampled independently from $\mathcal{N}(0, \Sigma)$

This can be solved exactly in 2-dimensions: without loss of generality suppose eigenvalues of $\Sigma$ are 1 and $k$, then

$$E[y^2] = \frac{k+1}{\left(\sqrt{k}+1\right)^2}$$

Is there a similarly elegant expression for $n$ dimensions?