This paper focuses on the structural properties of test channels, of Wyner's operational information rate distortion function (RDF), $\overline{R}(��_X)$, of a tuple of multivariate correlated, jointly independent and identically distributed Gaussian random variables (RVs), $\{X_t, Y_t\}_{t=1}^\infty$, $X_t: ��\rightarrow {\mathbb R}^{n_x}$, $Y_t: ��\rightarrow {\mathbb R}^{n_y}$, with average mean-square error at the decoder, $\frac{1}{n} {\bf E}\sum_{t=1}^n||X_t - \widehat{X}_t||^2\leq ��_X$, when $\{Y_t\}_{t=1}^\infty$ is the side information available to the decoder only. We construct optimal test channel realizations, which achieve the informational RDF, $\overline{R}(��_X) \triangleq\inf_{{\cal M}(��_X)} I(X;Z|Y)$, where ${\cal M}(��_X)$ is the set of auxiliary RVs $Z$ such that, ${\bf P}_{Z|X,Y}={\bf P}_{Z|X}$, $\widehat{X}=f(Y,Z)$, and ${\bf E}\{||X-\widehat{X}||^2\}\leq ��_X$. We show the fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF, $\overline{R}(��_X)$, satisfy conditional independence, $ {\bf P}_{X|\widehat{X}, Y, Z}={\bf P}_{X|\widehat{X},Y}={\bf P}_{X|\widehat{X}}, \hspace{.2in} {\bf E}\Big\{X\Big|\widehat{X}, Y, Z\Big\}={\bf E}\Big\{X\Big|\widehat{X}\Big\}=\widehat{X} $ and (2) similarly for the conditional RDF, ${R}_{X|Y}(��_X) \triangleq \inf_{{\bf P}_{\widehat{X}|X,Y}:{\bf E}\{||X-\widehat{X}||^2\} \leq ��_X} I(X; \widehat{X}|Y)$, when $\{Y_t\}_{t=1}^\infty$ is available to both the encoder and decoder, and the equality $\overline{R}(��_X)={R}_{X|Y}(��_X)$.