The following theorem was presented in my textbook without proof and I would be thankful if someone could refer me to a proof of it:
Suppose that $ \boldsymbol{X} \sim N(\boldsymbol{\mu},\boldsymbol{\Lambda}) $, where $\boldsymbol{\Lambda}$ can be partitioned as follows:
\begin{pmatrix} \boldsymbol{\Lambda_1} & \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{\Lambda_2} & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \vdots & \ddots & \vdots \\ \boldsymbol{0} & \boldsymbol{0} & \cdots & \boldsymbol{\Lambda_n} \end{pmatrix}
(possibly after reordering the components), where $\boldsymbol{\Lambda_1}, \dots, \boldsymbol{\Lambda_n}$ are matrices along the diagonal of $\boldsymbol{\Lambda}$. Then $\boldsymbol{X}$ can be partitioned into vectors $\boldsymbol{X^{(1)}},\boldsymbol{X^{(2)}},\dots,\boldsymbol{X^{(n)}}$ with $Cov(\boldsymbol{X^{(i)}})=\boldsymbol{\Lambda_i}$, $i=1,\dots,n$ in such a way that these random vectors are independent.