  A full coupled-cluster expansion suitable for sparse algebraic operations is
developed by expanding the commutators of the Baker-Campbell-Hausdorff series
explicitly for cluster operators in binary representations. A full
coupled-cluster reduction that is capable of providing very accurate solutions
of the many-body Schr\"odinger equation is then initiated employing screenings
to the projection manifold and commutator operations. The projection manifold
is iteratively updated through the single commutators $\left\langle \kappa
\right| [\hat H,\hat T]\left| 0 \right\rangle$ comprised of the primary
clusters $\hat T_{\lambda}$ with substantial contribution to the connectivity.
The operation of the commutators is further reduced by introducing a
correction, taking into account the so-called exclusion principle violating
terms, that provides fast and near-variational convergence in many cases.
