Thanks! That's what I thought because of the curves touching, but I wasn't sure if closing the loop made it okay.

Have you checked out two of Dyer's recent papers: 1) Embeddings of root systems. I. Root systems over commutative rings. J. Algebra 321 (2009), no. 11, 3226–3248. 2) Embeddings of root systems. II. Permutation root systems. J. Algebra 321 (2009), no. 3, 953–981. Not sure if you have seen those or if they are of interest, but thought I would mention them.

I will think a bit harder about this in a while, but I know that du Cloux did some interesting work that you may be interested in: emis.de/journals/EM/expmath/volumes/11/11.3/Cloux371_381.pdf