Two vectors $\mathbf{Y}$ and $\mathbf{Z}$ are uncorrelated if their cross-covariance matrix $\mathrm{cov}(\mathbf{Y}, \mathbf{Z})$ is a zero matrix.
Let $\mathbf{X}_1,\ldots,\mathbf{X}_n$ be $n$ sub-vectors of a normal random vector $\mathbf{X}$ i.e. $\mathbf{X} = (\mathbf{X}_1,\ldots,\mathbf{X}_n)^{\mathrm{T}}$.
Then will the following statement be true?
"Sub-vectors of a normal random vector are pairwise uncorrelated if and only if they are mutually independent"
I know that this statement is true for $n=2$ (proof here). I also know that this statement is true when all sub-vectors are scalars (i.e. random variables, see here). And it is obvious that independence implies uncorrelatedness. So can we generalise these facts as stated above?