  Borcherds-Kac-Moody algebras generalise finite-dimensional, simple Lie
algebras. Scheithauer showed that there are exactly ten Borcherds-Kac-Moody
algebras whose denominator identities are completely reflective automorphic
products of singular weight on lattices of square-free level. These belong to a
larger class of Borcherds-Kac-Moody (super)algebras Borcherds obtained by
twisting the denominator identity of the Fake Monster Lie algebra. Borcherds
asked whether these Lie (super)algebras admit natural constructions. For the
ten Lie algebras from the classification we give a positive answer to this
question, i.e. we prove that they can be realised uniformly as the BRST
cohomology of suitable vertex algebras.
