The drum spins fine during wash cycle. However, the agitator does not spin at all on my FEX831FS4 Frigidaire washer.
So, I removed the plastic agitator and inspected this rod on which the agitator goes on. The top of the rod can be spun freely by hand either direction.
Update to Jeff's answer
About solution #5. I believe the plastic agitator is fine. The teeth look square to me. Also the metal shaft does not appear to be spinning. I did experiment by putting a marker on shaft and it stayed in the same place. Here is a picture of agitator from the bottom:
About solution #2. I believe the belt is fine too. I recorded the belt spinning with high frame camera in both spin and agitate modes. The only difference I noticed was that the motor for each mode was spinning in the opposite directions.
Before getting to transmission I would like to rule out Solution #1. If I understand correctly then the drive block is the metal piece that sits between transmission shaft and plastic agitator. It somewhat wiggles up and down, but I noticed that if I turn drive block in either of directions then the thing drive block sits on (I assume transmission axle by peeking through the small gap?) turns together with it. Does this prove that drive block is fine too?
Does this confirm that the transmission is broken?