I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed very glancingly in the literature for years, but seemingly without anyone actually dealing with it carefully. One often feels that it can't be that such an basic thing has been ignored so long.

People in Lie theory have known for many decades that a basic and important concept in understanding Dynkin diagrams is "folding." My most basic question is thus:

Has anyone written a careful and systematic treatment of folding for arbitrary (valued) graphs, including the associated embeddings of Lie algebras?

I ask in large part because I need a fair amount of this theory for a paper I'm working on, and don't want to do too much wheel-reinventing or ignoring references that deserve mention. An incomplete list of what I have found:

- The closest thing I have found is this survey of Lemay, but it doesn't discuss the embeddings of algebras, since he works on the wrong side of the duality discussed by Humphreys in this question.
- I also found this paper of Berenstein and Greenstein useful, but it (and Kac's book, which it references) only deal with the finite type case.
- This paper of Naito has the ultimate result I'm looking for in terms of embeddings, but has no details (it's 4 pages long and contains one proof sketch) and no discussion of graphs.