The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices

We prove that every vector of $\mathrm{II}_{8n+1,1}$, with $n\ge1$, is a sum of two roots if and only if $n=1$ or $2$. This settles the Borisov--Nuer conjecture and implies the existence of Ulrich line bundles for every very ample polarization on a complex unnodal Enriques surface. For vectors of ne