Let $\mathbf{w}\sim \mathcal{N}(\mathbf{0}, \mathbf{\Sigma})$, and $\mathbf{A, B}$ are symmetric matrices, then it is known that
$ \operatorname{Cov}\left[\mathbf{w^TAw, w^TBw}\right]=2\operatorname{Tr}[\mathbf{A\Sigma B\Sigma}].$
The problem is to compute
$\operatorname{Cov}\left[\mathbf{w^TAw, v^TBv}\right]$, where $\mathbf{w, v}\sim \mathcal{N}(\mathbf{0}, \mathbf{\Sigma})$, and $\mathbf{A, B}$ are symmetric matrices.
Any leads would be appreciated.