[chuckm]

We are talking of energy. It doesn´t mater how God put energy inside water, wether by chemistry, thermochemitry, physic-chemestry or whatever means, it is energy.

It most certainly does matter! Thermal energy, as approximated by temperature, is present in any mass with a temp above 0 Kelvin. Whether that energy can be transferred usefully is a different matter. When I say "thermochemical energy", I am referring to the potential energy stored in a material that can be liberated, via a chemical reaction, in the form of heat.

You also wrote that hydrogen combinated with oxygen IS an energy source.
And what is water? From the best of my knoledge, it is a combination of two atoms of hydrogen with one atom of oxygen... or not?

If I phrased that poorly, I apologize. I am simply referring to the fact that hydrogen burns. When hydrogen burns, the chemical reaction (2*H2 + O2 -> 2*H20 + heat energy) produces heat as a by product. If you want to reverse that reaction (dissociate the water into H2 and O2), then you must add energy (2H2O ->electrolysis-> 2*H2 +O2).

You also wrote: "From a physical chemistry standpoint, that means the potential thermochemical energy of water is lower than that of oxygen and hydrogen, at the same pressure and temperature".

I don´t see how it is lower. When I use compression to burn diesel I don´t see why that compression energy will be higher than the energy produced by diesel combusion.

Again, hydrogen, in the presence of oxygen and an ignition source, will produce fire. Fire = energy release. Think of the situation like gravity: I'm standing on a building holding a bowling ball. At the moment I release that bowling ball, it has zero velocity. The kinetic energy of the ball is likewise zero
KE =1/2*m*v^2
However, it has a potential energy equal to the mass times the height times the acceleration due to gravity
PE = m*g*h.
At the moment the ball strikes the ground, its potential energy is zero (because the height equals zero). At the same time, the kinetic energy is at its highest because the velocity is at its highest.
Here's the key to the whole thing: the initial potential energy at the moment I dropped the ball is equal to the kinetic energy of the ball as it hits the pavement.
Now, I want to take the ball back up to the roof. I have to move the ball up there, meaning that I'm putting energy in (Work = Force * distance, where Force = mass * acceleration due to gravity). But due to inefficiency, the work (or change in energy) I expend in bringing the ball back up there is greater than the potential energy the ball acheives at the top of the building (think friction of the elevator, electrical losses in the motor, the energy I expend in walking both my mass and the bowling ball, etc).

NOW, getting back to the HHO system. Hydrogen is the bowling ball at the top of the building. Now exchange the bowling ball's initial potential for the energy stored in the chemical bonds (or if you prefer, the rest energy state of the electrons in a hydrogen molecule). Exchange the bowling ball's final kinetic energy for heat liberated during the chemical reaction. But producing the HHO is more like moving the bowling ball back on top of the building. It takes energy and it is not terribly efficient. Most HHO system run pretty freakin' hot (wasted energy). If you are drawing electrical current from the alternator, then you must also account for the alternator, the belt, and engine inefficiencies. Moving that bowling ball (dissociating water into hydrogen and oxygen) will take more energy than will be released when that same hydrogen is burned in the engine.

Figures are missing. And "movement is proved by moving", as the Greek said.

If you are right, figures will show it. Don't dispair.

If you want figures, I can provide them. If you take 4 grams of hydrogen and combine them with 32 grams of oxygen, forming 36 grams of water, you will release 118 kilocalories of heat. If you could reverse this reaction with 100% efficiency, it would take 118 kcal of energy to do so. But you will not be anywhere near 100% efficient. This is a good reference, though it does have a bunch of biology tacked on the latter part of the article.