  In this article, we describe the trace formulae of composition of several (up
to four) adjoint actions of elements of the Lie algebra of a vertex operator
algebra by using the Casimir elements. As an application, we give constraints
on the central charge and the dimension of the Lie algebra for vertex operator
algebras of class $\mathcal{S}^4$. In addition, we classify vertex operator
algebras of class $\mathcal{S}^4$ with minimal conformal weight one under some
assumptions.
