In an answer to a recent question on Worldbuilding, I suggested that an organism convert $\ce{H2O}$ into $\ce{H2O2}$. I suggested a few processes that yielded the desired final result ($\ce{2H2O \rightarrow H2O2 + H2}$ and $\ce{H2O + O3 \rightarrow H2O2 + O2}$). I noted that both were mere guesses. The feasibility of it was debated because hydrogen peroxide is a rocket fuel and contains a lot of energy (I heavily modified the answer and set it on an entirely different track after another mistake was pointed out).
Could some process similar to this ever happen in an organism?