  For an infinite chain bicomplex we show that the orthogonality and grading
conditions provide it with the structure of a bigraded differential algebra
with respect to a natural multiplication of several elements bicomplex spaces.
Corresponding bigraded algebra commutation relations generate a sequence of
non-vanishing cohomology invariants associated to vertex algebras. In
particular, we apply this result to the bicomplex of grading-restricted vertex
algebra cohomology endowed with a multiplication we introduce. We provide
examples associated to various choices of vertex algebra bicomplex subspaces.
The generators and commutation relations of the bigraded differential algebra
form a continual Lie algebra with the root space provided by a
grading-restricted vertex algebra.
