Could we add something to jet fuel to aerosolize it in the atmosphere and cool off the planet?

Short answer: No
Long answer: That would require incomplete combustion, which would drastically reduce fuel efficiency. Not to mention, all of the possible undesired compounds that would likely be created from the combustion of fuel with those additives. Also, additives tend to foul fuel nozzles, which requires more maintenance and more frequent engine cleaning.
Planetary heating is caused more by greenhouse gasses reflecting and trapping heat, rather than letting the Earth radiate that heat into space. The best strategy with current technology is to reduce the creation of those greenhouse gasses.

Do you want “chemtrails”? Because that’s how you get “chemtrails”.

Potentially? My concern is, how many living things are you going to poison to death in the process of injecting aerosolized poison into the atmosphere?

I've listened to a podcast about geoengineering, but it's unfortunately in Swedish. The name of the podcast? P3 Dystopia.

Yeah... Geoengineering might sound good on paper, but often comes with all kind of terrible side-effects.

If it were harmless it wouldn't do anything.

Generally when you are combusting fossil fuels like that, the fuel itself is consumed in the process. You tend to get CO2+H2O in a vapor trail. Partial combustion may show up as soot or carbon monoxide, but most people probably don't want planes to be belching black smoke and poisonous gas.

Generally when you are combusting fossil fuels like that, the fuel itself is consumed in the process. You tend to get CO2+H2O in a vapor trail. Partial combustion may show up as soot or carbon monoxide, but most people probably don't want planes to be belching black smoke and poisonous gas.