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You'd get most of the energy back when the compressed air decompresses during the
"power" cycle ("power" is in quotes because with the injectors off, there's no added power...just a decompression). True, there is a bit of loss due to heat loss during compression (imperfect adiabaticity), but that's probably less than friction loss and pumping loss (moving air through the engine as it rotates).
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Daily driver:
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