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This paper addresses ignition histories of diffusion flames in unstrained hydrogen-air mixing layers for initial conditions of temperature and pressure that place the system below the crossover temperature associated with the second explosion limit of hydrogen&-oxygen mixtures. It is seen that a two-step reduced chemical-kinetic mechanism involving as main species H₂, O₂, H₂O, and H₂O₂, derived previously from a detailed mechanism by assuming all radicals to follow a steady-state approximation, suffices to describe accurately the ignition process. The strong temperature sensitivity of the corresponding overall rates enables activation-energy asymptotics to be employed for the analysis, following the ideas developed for mixing-layer ignition by Liñán and Crespo in 1976 on the basis of one-step Arrhenius model chemistry. When the initial temperatures of both reactants differ by a relative amount that is of the order of or smaller than the ratio of this temperature to the effective activation temperature, the chemical reaction is seen to occur at a significant rate all across the mixing layer. The ignition time is then determined as a thermal runaway in a parabolic problem describing the evolution of the temperature increment and the H₂O₂ concentration, with local accumulation, chemical reaction, and transverse convection and diffusion, all being important. By way of contrast, when the air side is sufficiently hotter than the hydrogen side, as often occurs in applications, ignition occurs in a thin layer close to the air-side boundary, enabling a simplified description to be developed in which the ignition time is determined by analyzing the existence of solutions to a two-point boundary-value problem involving quasi-steady diffusion&-reaction ordinary differential equations.