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