Technically it's not really a violation though, the water is a fuel source and a consumable. It's kinda like using natural gas to power a pump that pumps natural gas into the system. But the main question is whether or not there is enough energy in the water to to extract said energy from more water. I'm thinking there isin't, or we'd see this concept used everywhere, unless there is a patent on it, that is.
Uhhh, technically it's a huge violation.
Out of curiosity, what do you think becomes of the hydrogen?
Let me summarize what you think is going on:
"Hey, if you mix hydrogen and oxygen, you make water, and it gives off a LOT of energy! So, what we'll do is start out with some energy, separate the hydrogen from the water. Then get a lot of energy when we recombine them!"
You may be thinking, "but what if we combine the hydrogen with something that gives off even more energy than when it combines with water!" Then, that substance would be even more stable than water. <looks around> There's a lot of water on this planet. If you had something, say a substance with one carbon atom and 4 hydrogen atoms. If you react it with oxygen, you'll wind up with carbon dioxide, water, and a bunch of energy. Thus, this is a lower energy state.
In what I proposed above - producing hydrogen with the gallium - "theoretically" the amount of energy you get from the hydrogen would be equal to the amount of energy required to split the aluminum atom from the aluminum oxide to make the aluminum in the first place. That's not theoretically what happens though - entropy is a bitch. No matter what, you get less energy out than you put in. In this case, if you had a source of aluminum, that would be your fuel. But, aluminum is not a fuel source - it takes a fuel source to make the aluminum, and your raw material is aluminum oxide. (Actually, the raw material is bauxite which contains hydrated aluminum along with some iron oxides.)