6-transposition property of τ-involutions of vertex operator algebras

In this paper, we study the subalgebra generated by two Ising vectors in the Griess algebra of a vertex operator algebra. We show that the structure of it is uniquely determined by some inner products of Ising vectors. We prove that the order of the product of two $\tau$-involutions is less than or equal to 6 and we determine the inner product of two Ising vectors.