Hi, new here.
I have an '87 450 rebel that I recently got starting after cleaning the carbs. Engine starts very easy when cold. However after about 20 seconds of running, I can feel the left cylinder get way hotter than it should get. Meanwhile the right cylinder is completely normal.
About 30 seconds in, the engine dies and is now harder to start. Have to hold the throttle or else it will die. After I shut the engine down, I can see white/gray smoke coming from the left exhaust pipe. Also smells like bad burning right next to the cooling fins. I adjusted the valves, and there are not any apparent air leaks on any particular side. New synthetic oil BUT didn't change the oil filter. Bike has 10k miles.
Sounds like the carb is not completely clean. Did you do yourself? Were you able to use an ultrasonic cleaner? Any little bits left in one of the many ports in there could mean trouble.
Did you check compression? Are the carb vacuum lines intact? They can get dry rot (after+30 years of service!) along with other seals, so should be checked very closely. I suggest replacing them with new rubber, guaranteed not to be leaking then!
How long did the bike not run? New oil, but old filter? Now you have dirty new oil that likely can't get through an already 'clogged' filter. I would suggest changing the filter when you change the oil, but that does not sound like the issue here.
I think your (carb on the right) is not functioning properly. I highly suggest you use JRW for carb rebuild. I use him as I do not have an ultrasonic cleaner and simply would not do it as thorough as he does and he also test them before sending back, so it is bolt-on and go.
Sounds like the left side is starved for oil lube. Maybe the oil pressure relief valve is stuck open? It's located on the right side of the engine case at the bottom of the sump, under the chain driven oil pump. The valve was made for disassembly and cleaning.
My best guess...
On a side note, I would have replaced the oil filter, especially if you don't know the history of the bike.
The oil pressure sensor is called an oil pressure switch. My manual doesn't say at what pressure the switch would would close, but if one side is getting proper lube that might be just enough pressure to keep it from closing. The oil pressure relief valve lifts between 55-74 PSI (382-510 kPa). If it's stuck open then the oil would follow the path of least resistance and go to the right side and and possibly not send enough oil to the left side which is farther away from the pump.
My manual does say to check it for proper operation. Disassemble and look for scoring inside. It is spring loaded, so be wary of flying springs!