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Herein we study conformal vectors of a Z-graded vertex algebra of (strong)CFT type. We prove that the full vertex algebra automorphism group transitivelyacts on the set of the conformal vectors of strong CFT type if the vertexalgebra is simple. The statement is equivalent to the uniqueness of self-dualvertex operator algebra structures of a simple vertex algebra. As anapplication, we show that the full vertex algebra automorphism group of asimple vertex operator algebra of strong CFT type uniquely decomposes into theproduct of certain two subgroups and the vertex operator algebra automorphismgroup. Furthermore, we prove that the full vertex algebra automorphism group ofthe moonshine module over the field of real numbers is the Monster.

On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group