This paper extends past analyses of liquid-oxygen-hydrogen flames at supercritical pressure by providing quantitative results that characterize multicomponent diffusion processes in the flame zone of a shear-coaxial injector element. High-fidelity simulations, using both the large-eddy-simulation and direct-numerical-simulation techniques, have been performed using detailed treatments of thermodynamic, transport and chemical kinetics. Results are presented for a condition where oxygen is injected in a cryogenic state, at a subcritical temperature and supercritical pressure, and hydrogen is injected in a supercritical state. This condition has significant technical relevance in liquid-rocket engines but is not well understood. For this situation a diffusion dominated mode of combustion occurs in the presence of exceedingly large thermo-physical property gradients. The flame anchors itself in the interfacial region of high shear that exists between the liquid-oxygen core and the annular hydrogen jet. Intense property gradients approach the behavior of a contact discontinuity in this region. Significant real-gas effects and transport anomalies coexist locally in colder regions of the flow, with ideal gas and transport processes occurring within the flame zone. The current work provides a detailed analysis focused on the effects and relative contributions of various diffusion mechanisms and the net coupled effect of these processes on the observed mode of combustion.