Posted by: @narnian78

I guess Lewis thought of the magic rings in Uncle Andrew’s study later since the Lefay Fragment does not have them.

The Lefay Fragment doesn't have Uncle Andrew either (only Aunt Gertrude, who is a thoroughly unpleasant character, but definitely not the sort who would meddle around with magic). I don't think we have any clear indication of when Lewis thought up the magic rings, or the Wood between the Worlds, but as they're unique elements of The Magician's Nephew and there aren't even any hints of them in the books that he wrote earlier, I would guess Lewis didn't come up with those concepts until about the time when he started writing MN, which, as far as we know from Hooper, he finished writing some time in 1952.

Since Narnia is not mentioned in the fragment it may not have occurred to him at that time to make it into an another Narnia story.

I really do now think that's the case, absolutely.

Indeed, that blog by Brenton Dickieson ("A Pilgrim in Narnia") that I linked to earlier may also be somewhat off the mark, with the author's assertion that once Lewis had written LWW, "Aslan pulled the whole story together — and then the whole series", which, Dickieson concludes, must mean that Lewis, from there on in, never had any serious thoughts of writing any other stories for children except ones set in Narnia. Hence Dickieson's assertion that the Lefay Fragment was almost certainly written before LWW and before Lewis had any concepts of Narnia and Aslan in his head.

But being reminded of the "Plots List" (thanks to re-reading the part of Past Watchful Dragons that deals with it and with the Lefay Fragment) has made it stand out all the more that the first plot in that list, the "Ship" one, was actually — as it stands — a non-Narnia story that Lewis was toying with AFTER he'd written LWW. As I said before, of course he eventually returned to that basic idea of a ship sailing to islands and recycled it for the third Narnia adventure. But in its original form, that plot really could not have been for a Narnia book, much though Hooper mistakes it for one by assuming that the four "plots" that Lewis jotted down hastily in that notebook — just because they conclude with "Sequel to L.W.W." — are all in fact one plot for one future Narnia story.

The "Ship" story, of two children travelling back in time to islands that "have not existed for millennia", implicitly occurs within our world. Lewis suggests "Various islands (of Odyssey and St Brendan) can be thrown in" — those are islands from our own world's mythologies, the journey of Odysseus in ancient Greek legend and the myth of St Brendan's Island in the Atlantic (which was supposedly discovered by the Irish missionary St Brendan and included on a number of medieval maps, including the Mappa Mundi of Hereford, from which Lewis almost certainly got the idea of the Dufflepuds!). There's absolutely no sense in that sketch of a story that this journey back in time has any connection with Narnia; Lewis imagined the quest as being to save a sick king who can only be healed by the blood of a boy from the far future, not the blood of a boy from another world, as would need to be the case if Lewis was intending this to be another Narnia story. The king must be a king from somewhere in the (mythical) past of this world, not Narnia. There's no other way that story could work.

I'm harping on about this simply because it's solid evidence that Lewis WAS thinking up ideas for stories unrelated to Narnia even after he'd finished writing the first Narnia book and started considering the second. Which means he very well could have written the Lefay Fragment directly after he completed LWW (as Lancelyn Green and Hooper assume), with absolutely no intention of this story being a Narnia story. And that really does explain everything about the Lefay Fragment perfectly well, as far as I can see.

 The next book Prince Caspian seem more like it was intended to be a second Narnia book since it has a portal into that world, the train station.  He may have realized that he couldn’t do without something to transport his characters into another world so he didn’t use the fragment and created a different story.

Prince Caspian was always definitely meant to be the second Narnia story — we can unarguably see its basis in the last item on the Plots List, where it's headed "Sequel to L.W.W." (I've already quoted most of it above). That must have been the idea from that list that Lewis decided to go with fully, and since it was specifically a sequel to the book he'd already written about Narnia, it's natural that he started it with the same four child characters (the same opening sentence, actually!) as the first book. And since he'd already had the Professor assert that they wouldn't get back into Narnia by the wardrobe (convenient plot twist to forestall any readers asking "So couldn't they have tried to go back through the wardrobe again before they went home from the Professor's house??), there needed to be a different way of getting them there this time. Perhaps it occurred to Lewis that he'd barely used Susan's magic horn as a plot element in LWW — she does blow it once, but only to call Peter, who is within earshot anyway, so there's nothing particularly magical about that. But if she'd left it behind somewhere in Narnia, and if somebody in Narnia's future found it and blew it at a time of great need...

Anyway, it does seem that once Lewis had got stuck into writing that second Narnia book, then he never entertained serious thoughts of writing any other children's stories, apart from more Narnia Chronicles, from there on in. But in those months between finishing LWW and beginning PC, he clearly did have a few intriguing ideas floating around in his head for fantasy stories set in our world, rather than in Narnia. I do agree with Dickieson's hypothesis that the reason he didn't get anywhere with those other plots was because something about Narnia — I, and most of us here, would say Aslan himself — took hold of him and pulled his creative thoughts deeper into Narnia and away from any other directions. That explains why he never got further with either the Lefay plot (a boy in England who can talk to animals and trees) or the original Ship plot (a boy whose blood somehow has the power to heal a king in the mythical past). The real magic was in Narnia, and in Aslan, and Lewis must have realised increasingly that that's what he needed to stick with — though he did salvage a few good ideas from those discarded earlier stories and weave them into Narnia along the way.

The creation idea probably came later, although he may have had it long before writing the sixth book. 

True, we still don't know exactly when Lewis had the idea of writing the story of Narnia's creation, although I think it's now safe to say for sure that it wasn't while he was writing the Lefay Fragment. Unfortunately we have so few surviving drafts and notes from his entire creative process, and even people who knew him could only guess about these things, not always accurately, so I doubt we'll ever know for sure. But it probably doesn't matter. I'm just glad he wrote the stories he did, and that all seven of the Chronicles, as we now know them, are as good as they are!