  The Gray and Wyner lossy source coding for a simple network for sources that
generate a tuple of jointly 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 distortion at the two decoders is
re-examined using (1) Hotelling's geometric approach of Gaussian RVs-the
canonical variable form, and (2) van Putten's and van Schuppen's
parametrization of joint distributions ${\bf P}_{X_1, X_2, W}$ by Gaussian RVs
$W : \Omega \rightarrow {\mathbb R}^n $ which make $(X_1,X_2)$ conditionally
independent, and the weak stochastic realization of $(X_1, X_2)$. Item (2) is
used to parametrize the lossy rate region of the Gray and Wyner source coding
problem for joint decoding with mean-square error distortions ${\bf
E}\big\{||X_i-\hat{X}_i||_{{\mathbb R}^{p_i}}^2 \big\}\leq \Delta_i \in
[0,\infty], i=1,2$, by the covariance matrix of RV $W$. From this then follows
Wyner's common information $C_W(X_1,X_2)$ (information definition) is achieved
by $W$ with identity covariance matrix, while a formula for Wyner's lossy
common information (operational definition) is derived, given by
$C_{WL}(X_1,X_2)=C_W(X_1,X_2)
  = \frac{1}{2} \sum_{j=1}^n
  \ln
  \left(
  \frac{1+d_j}{1-d_j}
  \right),$ for the distortion region $ 0\leq \Delta_1 \leq
\sum_{j=1}^n(1-d_j)$, $0\leq \Delta_2 \leq \sum_{j=1}^n(1-d_j)$, and where $1 >
d_1 \geq d_2 \geq \ldots \geq d_n>0$ in $(0,1)$ are {\em the canonical
correlation coefficients} computed from the canonical variable form of the
tuple $(X_1, X_2)$. The methods are of fundamental importance to other problems
of multi-user communication, where conditional independence is imposed as a
constraint.
