  In this paper, the rate-distortion theory of the Gray-Wyner lossy source
coding system is investigated. For the case of jointly Gaussian distributed
sources, we establish an expression for the rate-distortion function under the
constraint of quadratic distortion. Using the proposed rate-distortion
function, any corner point on the rate-distortion region can be conveniently
calculated. We take Wyner's common information as an example and provide a
general and simple method to solve this problem. Through the analysis of the
rate-distortion function, the rate on each layer and covariance matrix of
auxiliary random variables and the sources are also presented in this paper.
