Cross-covariance matrix calculation based on known covariance matrices

Question

I have a very common problem, but couldn't find any proper evidence why it isn't possible (with proof) or how to do it (with a method). The problem is that 2 random vectors are modeled with white gaussian distribution and their covariance matrices are known: $P_{xx}$ and $P_{yy}$. A $3^{rd}$ vector can be expressed with these vectors and the task is to calculate its covariance matrix. Using the definition of the covariance matrix the following equation stands for $E_{zz}$: \begin{equation} E\{ z^Tz\}= \begin{bmatrix}A & B\end{bmatrix} \begin{bmatrix}E_{xx} & E_{xy} \\ E_{yx} & E_{yy}\end{bmatrix} \begin{bmatrix}A^T \\ B^T \end{bmatrix} \end{equation} This requires the knowledge of $E_{xy}=(E_{yx})^T$. On this link: Cross-Covariance matrix from two covariance matrices someone told that it is not possible to calculate $E_{xy}$ without knowing x and y joint distribution. If that's correct what is the best way to approximate $E_{zz}$?