Vertex operator algebra with two Miyamoto involutions generating S3

In this article we study a VOA with two Miyamoto involutions generating S3. In [math.GR/0112031], Miyamoto showed that a VOA generated by two conformal vectors whose Miyamoto involutions generate an automorphism group isomorphic to S3 is isomorphic to one of the four candidates he listed. We construct one of them and prove that our VOA is actually the same as VA(e,f) studied by Miyamoto. We also show that there is an embedding into the moonshine VOA. Using our VOA, we can define the 3A-triality of the Monster.