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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.

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  • $\begingroup$ Are the "exceptional types" for $E_8$ made unique by requiring that they embed primitively? For example, $A_7$ can embed either as an index-$2$ sublattice of the unique $E_7$, or as the orthogonal complement of a primitive vector of norm $8$; likewise $A_3^2$, $A_5 \oplus A_1$, $A_3 \oplus A_1^2$, and $A_1^4$ can be either primitive or contained with index $2$ in $D_6, E_6, D_5, D_4$ respectively. (For the $D_6$ and $D_5$ sublattices this can be seen more easily by writing $A_3^2 = D_3^2$ and $A_3 \oplus A_1^2 = D_3 \oplus D_2$.) $\endgroup$ May 3, 2021 at 14:18
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    $\begingroup$ Yes, for each of the exceptional types there exist exactly one primitive and one imprimitive embedding. $\endgroup$ May 3, 2021 at 14:34

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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 $E_{10}$.

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