I should have mentioned the nut is a self locking shoulder nut, so think the split washer is not required. (Sorry, I get a bit retentive, being a software engineer and have been wrenching on autos and motorcycles for fifty years )
#4
StarTech
Idler pulley are usually mounted using smooth flange nuts. In this case they are 1/2-13 nylon flange lock nuts and should be torque to 30 ft-lbs minimum per screw torque charts. I can't tell what grade they are from the images so I would go with the grade 2 specs to be safe. They use flange as the built-in catches the inner bearing hub making for an additional washer unnecessary. Also a spilt washer might damage the bearing seal if too large of one is used. Besides these pulley should have a pulley shield install which is modified flat washer.
I use flange nuts and screws where ever space allows as there is no need a separate washer. They come both smooth flange and serrated (lock style) , although the serrated vesion damage the mating surface which why nylon lock version is preferred over the serrated except extreme vibration areas.
Idler pulley are usually mounted using smooth flange nuts. In this case they are 1/2-13 nylon flange lock nuts and should be torque to 30 ft-lbs minimum per screw torque charts.