  For the double complex structure of grading-restricted vertex algebra
cohomology defined in \cite{Huang}, we introduce a multiplication of elements
of double complex spaces. We show that the orthogonality and bi-grading
conditions applied on double complex spaces, provide in relation among mappings
and actions of co-boundary operators. Thus, we endow the double complex spaces
with structure of bi-graded differential algebra. We then introduce the simples
cohomology classes for a grading-restricted vertex algebra, and show their
independence on the choice of mappings from double complex spaces. We prove
that its cohomology class does not depend on mappings representing of the
double complex spaces. Finally, we show that the orthogonality relations
together with the bi-grading condition bring about generators and commutation
relations for a continual Lie algebra.
