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We generalize Conway-Sloane's constructions of the Leech lattice fromNiemeier lattices using Lorentzian lattice to holomorphic vertex operatoralgebras (VOA) of central charge 24. It provides a tool for analyzing thestructures and relations among holomorphic VOAs related by orbifoldconstruction. In particular, we are able to get some useful information aboutcertain lattice subVOAs associated with the Cartan subalgebra of the weight oneLie algebra. We also obtain a relatively elementary proof that any stronglyregular holomorphic VOA of central charge $24$ with a non-trivial weight onesubspace can be constructed directly by a single orbifold construction from theLeech lattice VOA without using a dimension formula.

Orbifold construction and Lorentzian construction of Leech lattice vertex operator algebra