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.