As stated in the title, I need an expression for the covariance between the elements of a vector $X \in \mathbb{R}^n \sim \mathcal{N}(\mu, \Sigma)$, and its norm $\lVert X \rVert$.
There is a formula for the covariance between $X$ and $\lVert X \rVert^2$, as I wrote in the answer to another question, which seems to be an easier case due to the lack of the square root, but I can't find anything for $\lVert X \rVert$. I'm not sure what approach I could take to try to derive such an expression myself. Any help would be greatly appreciated.