Classification of the vertex operator algebras VL+ of class S4S4

In this paper, we prove that an even rootless lattice L   is isomorphic to 2A12A1, 2D4, 2E8, or BW16BW16 if the lattice type vertex operator algebra VL+ is of class S4S4. In addition, we prove that the vertex operator algebra V2D4+ is of class S5S5, and the vertex operator algebras V2A1+, V2E8+, and VBW16+ are of class S7S7.