  Examined in this paper, is the Gray and Wyner achievable lossy rate region
for a tuple of correlated multivariate Gaussian random variables (RVs) $X_1 :
\Omega \rightarrow {\mathbb R}^{p_1}$ and $X_2 : \Omega \rightarrow {\mathbb
R}^{p_2}$ with respect to square-error distortions at the two decoders. It is
shown that among all joint distributions induced by a triple of RVs $(X_1,X_2,
W)$, such that $W : \Omega \rightarrow {\mathbb W} $ is the auxiliary RV taking
continuous, countable, or finite values, the Gray and Wyner achievable rate
region is characterized by jointly Gaussian RVs $(X_1,X_2, W)$ such that $W $
is an $n$-dimensional Gaussian RV. It then follows that the achievable rate
region is parametrized by the three conditional covariances $Q_{X_1,X_2|W},
Q_{X_1|W}, Q_{X_2|W}$ of the jointly Gaussian RVs. Furthermore, if the RV $W$
makes $X_1$ and $X_2$ conditionally independent, then the corresponding subset
of the achievable rate region, is simpler, and parametrized by only the two
conditional covariances $Q_{X_1|W}, Q_{X_2|W}$. The paper also includes the
characterization of the Pangloss plane of the Gray-Wyner rate region along with
the characterizations of the corresponding rate distortion functions, their
test-channel distributions, and structural properties of the realizations which
induce these distributions.
