Nimbostratus
Nimbostratus"the server in Pool 2 to initiate a connection to another load-balanced virtual server on the LTM and have it maintain session state?"
Indeed. The application that lives on Pool2:AllNodes knows that it will want to connect to Pool1, and it will need to connect to the Node:Session that the user has previously initiated.
My understanding is that once the user connects to Pool1, the returned header will include a cookie from the LB, detailing the pool:node combo, and when the user submits that cookie back to the LB, the user will then be directed to the same node of the pool, but the cookie will not leave the LB (the application on Node1 never sees the cookie).
I guess what I'm trying to figure out is how does this session access function if there is only 1 server in each "Pool" and LTM is not involved (ie no server load balancing).
Nodes A & B are both members of Pool1, Nodes Y & Z are both members of Pool2, so both pools are load balanced. I apologize if the diagram didn't make that clear.