  We establish the rate region of an extended Gray-Wyner system for 2-DMS
$(X,Y)$ with two additional decoders having complementary causal side
information. This extension is interesting because in addition to the
operationally significant extreme points of the Gray-Wyner rate region, which
include Wyner's common information, G{\'a}cs-K{\"o}rner common information and
information bottleneck, the rate region for the extended system also includes
the K{\"o}rner graph entropy, the privacy funnel and excess functional
information, as well as three new quantities of potential interest, as extreme
points. To simplify the investigation of the 5-dimensional rate region of the
extended Gray-Wyner system, we establish an equivalence of this region to a
3-dimensional mutual information region that consists of the set of all triples
of the form $(I(X;U),\,I(Y;U),\,I(X,Y;U))$ for some $p_{U|X,Y}$. We further
show that projections of this mutual information region yield the rate regions
for many settings involving a 2-DMS, including lossless source coding with
causal side information, distributed channel synthesis, and lossless source
coding with a helper.
