The F800GS does not have a secondary air injection system.
The hose you point to (#4) in your above post is in fact the crankcase breather hose that feeds into the throttle bodies. This is not a secondary air injection system. Basically what #4 is doing is that all of the gases from the crankcase are getting fed to the throttle bodies to be burned with the rest of the mix. This controls emissions of crankcase gases into the atmosphere. There are many ways a manufacturer can control emissions to get their bikes within EPA and Euro spec without the use of a secondary air injection system. This crankcase breather feed system is one of them. Another is the charcoal canister we find on the US (and some other country's) bikes. With this system, a vent hose from the gas tank is fed to a charcoal canister which stores the vented gasoline vapors. Then when the engine is started the stored vapors get sucked into the throttle bodies and burned with the rest of the mix. BMW uses both of these methods on the F650GS2 and F800GS, but not the secondary air injection system.