The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) X1:Ω→ℝp1 and X2:Ω→ℝp2, 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 PX1,X2,W by Gaussian RVs W:Ω→ℝn which make (X1,X2) conditionally independent, and the weak stochastic realization of (X1,X2). 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 E{||Xi−X̂i||2ℝpi}≤Δi∈[0,∞],i=1,2, by the covariance matrix of RV W. From this then follows Wyner's common information CW(X1,X2) (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 CWL(X1,X2)=CW(X1,X2)=12∑nj=1ln(1+dj1−dj), for the distortion region 0≤Δ1≤∑nj=1(1−dj), 0≤Δ2≤∑nj=1(1−dj), and where 1>d1≥d2≥…≥dn>0 in (0,1) are {\em the canonical correlation coefficients} computed from the canonical variable form of the tuple (X1,X2). The methods are of fundamental importance to other problems of multi-user communication, where conditional independence is imposed as a constraint.