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That would be a perpetual motion machine, and violate the first law of thermodynamics.
The amount of energy you get by burning hydrogen (creating water) is exactly the same as you spent to split water into hydrogen and oxygen.
assuming no losses
Which in practice there absolutely would be.
No it's not. They're referring to exactly what a car or fusion does. To "break even"( in fusion terms), you must produce more energy than is being put in to maintain it. In a car, you turn some of that combustion power back into electrical power via the alternator and recharge the battery that you used to start the car.
They're just asking if the same principle can apply: using a quick burst of auxillary power to get it going that you then recoup from the excess power created by the hydrogen combustion. And keep in mind, you ARE creating excess power. It's what moves the vehicle lol.
You don't create power; you convert it and harness it. In an ICE, you convert chemical energy (gas or similar) into kinetic energy (explosion, turning a crankshaft, and rotating the wheels. Plus some of it going into the alternator) and heat, with a considerable amount still left as unused chemical energy (largely in the form of exhaust/soot)
If you separate parts of the process (such as splitting water into hydrogen) the pieces you are looking at (burning hydrogen as fuel) could be very useful. You're still converting chemical energy into kinetic energy and heat, and that may (or may not) be a better system than carrying around an electric battery with that same amount of energy.
There is simply no way to start with water, perform a series of self-contained chemical reactions ending back with water, and having more energy than you started with.
You're talking about a spark plug. Water doesn't magically split itself apart, that binding energy has to come from somewhere. Fusion still consumes fuel and generates exhaust.