A class of vertex operator algebras with building blocks L(cm,0)

Let [Formula: see text] be a moonshine-type vertex operator algebra generated by Virasoro vectors [Formula: see text] of central charge [Formula: see text], where [Formula: see text], such that the Virasoro vertex operator algebra generated by [Formula: see text] is simple, [Formula: see text], and [Formula: see text] for distinct [Formula: see text]. We prove that if [Formula: see text] is simple, then [Formula: see text] is uniquely determined by its Griess algebra [Formula: see text]. If [Formula: see text], we further prove that [Formula: see text] is simple. As a result, we deduce that the subalgebra [Formula: see text] of the parafermion vertex algebra [Formula: see text] generated by [Formula: see text], for [Formula: see text], is simple and isomorphic to [Formula: see text], where [Formula: see text] is the root system of [Formula: see text].