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SPOILERS:
Gabe doesn't get a name until the very end of the play. I thought that things were handled really well with him actually. They didn't make it inherently obvious in the first quarter of the play that Diana is the only one that can see or hear him. He was always conveniently not around whenever anybody else was. Or if he was around, in the same "room", he wasn't really interacting with anybody.
Gabe was the son of Diana and Dan and he died when he was only eight months old. Diana never really accepted this. It was neat in the play that they very slowly revealed details of him, first that he had died, and then that he had died before Natalie was born, and then that he was only eight months old, and then what he died of.
He was not helpful during therapy, because the end goal was to have her stop seeing the ghost/memory of her son, and he didn't want to go. He had this interesting song: "I'm Alive" that explained who he was, part ghost, part memory, part Diana's wishful thinking.
Diana decides that she's finally ready to say good-bye to him, when he sings this song: "There's a World", about how he knows a place where they can be together forever. They dance around on the stage and then exit.
The lights come down and then go up to Madden, making notes. Diana is in the hospital. She was found unconscious at home, with slash marks on her wrists. That part was… I must say… very creepy and very well done. Certainly gave that rather pretty song a new meaning.
Diana has ECT and forgets the past eighteen or so years of her life, and her husband and daughter are helping her to remember, but they don't mention their son. She remembers him anyway, and has a rather sane conversation about what happened to the baby. Dan is worried but it seems to be going okay, until she sees Gabriel. This time, she knows what that means, and she gets her daughter to drive her off to see the doctor.
Eventually, Diana decides that all the support from her husband is actually not helping her, and she sees now that it's killing him. She decides to leave. I think that she views it as setting Dan free, but he doesn't see it that way. Earlier, he sang a song to Diana about how he always leaves one light on in the house for her, but when she leaves, he sits there in the dark.
When she leaves, she leaves Dan and Gabe behind. Dan starts the reprise of "I Am The One" and Gabe joins in, and then Dan says, "Go away."
Gabriel grabs onto him and says that he won't go. Dan finally looks at him and gives him his name. This was not the last scene of the play, after that song Natalie came home and turned the light on, and they went into the finale song, but it could very well have been.