The Lufthansa Senator Lounge and Lufthansa Business Lounge are both operated by Lufthansa, a Star Alliance member.
The King David Lounge is operated by EL AL and exclusive for EL AL. LY isn't in *A.
The Dan Lounge is what's called a "third-party contract lounge" and is there for all the airlines in TLV that don't operate their own lounge (i.e. LY only).
So *G flying *A, or *A First/Business class, as well as OW Emerald/Sapphire, OW First/Business, ST Elite Plus and ST First/Business pax all have access to the Dan only