Classification of root lattice embeddings in $E_{10}$ There is a well-known classification result which gives a complete list of all root lattices which can embed into the lattice $E_8$. Moreover, each of these root lattices admits a unique embedding modulo the action of the Weyl group $W(E_8)$ except for five exceptional types:
$$A_7, \; A_3^2, \; A_5 \oplus A_1,\; A_3 \oplus A_1^2,\; A_1^4$$
I would be very interested to know whether a similar result is known for root lattice embeddings into the even unimodular lattice $E_{10}=E_8 \oplus U$ (where $U$ is the hyperbolic plane), and in particular how to count the number of inequivalent embeddings for a given root lattice.
EDIT: Is there any hope to solving this by playing around with the Dynkin diagram for $E_{10}$? My main concern is that $E_{10}$ does not arise from a semisimple Lie algebra, however Friedman - On the geometry of anticanonical pairs (p.83, Example 9.21) seems to imply that a classification should be possible, but unfortunately doesn't give any details.
 A: The results I am familiar with in this direction are in the related area of regular subalgebras, or $\pi$-systems, as in Carbone et. al. here. There, they give a good amount of information about the possible subalgebras in Kac--Moody algebras (and the overextended cases like $E_{10}$) coming from an embedding of compatible root spaces; they also look at possible Weyl group orbits of these subsystems and the corresponding embeddings. The references within include a lot of what I have seen in this area.
A paper not in the references of Carbone et. al. by Felikson and Tumarkin, here, looks at the specific case of hyperbolic regular subalgebras of hyperbolic Kac--Moody algebras, classifying them by subgroups of the Weyl group. An interesting question posed in their paper (and might remain open, I am not sure) is whether, for an arbitrary symmetrizable Kac--Moody algebra, any reflection subgroup of W should correspond to a regular subalgebra embedding.
In terms of determining embeddings from the Dynkin diagram, I would be surprised if one could make such an algorithm. Viswanath showed here that any hyperbolic Kac--Moody algebra with a symmetric Cartan matrix can be embedded as a regular subalgebra into $E_{10}$ by explicitly constructing the $\pi$-system of roots; since these would include Dynkin diagrams with multiple lacings, I wouldn't expect a nice way to read this information off from the diagram for $E_{10}$.
