Could a person use a device that splits water like for hydrogen fuel cell car but keep oxygen for scuba diving?

I heard that this canadian invented a device that splits water into hydrogen and oxygen. Companies use this device to store hydrogen in tanks until it can be used for a hydrogen fuel cell car(like they do in iceland). My question is can you store the Oxygen instead to go on the ultimate scuba adventure? My dads a scientist, but we’re not talking right now so I thought I would ask here.

Hydrogen-powered jet planes?

Background information:

1. Space shuttle's main engines (not the boosters) are hydrogen-powered.

2. There are (experimental) hydrogen-powered cars which actually burn hydrogen instead of gasoline (I'm not talking fuel cells, but hydrogen-based combustion engines!).

Considering that, would it be possible to have jet planes using hydrogen instead of kerosene? Would the re-design costs for the engines be prohibitively high? Would keeping the hydrogen liquid be too expensive at the airports and maybe impossible in the plane tanks? Would it be too dangerous, because of hydrogen's high inflammability?

If not, why is no-one working on that? Why is there development on cleaner cars, but not airplanes?