  We construct a projection-based cluster-additive transformation that
block-diagonalizes wide classes of lattice Hamiltonians
$\mathcal{H}=\mathcal{H}_0 +V$. Its cluster additivity is an essential
ingredient to set up perturbative or non-perturbative linked-cluster expansions
for degenerate excitation subspaces of $\mathcal{H}_0$. Our transformation
generalizes the minimal transformation known amongst others under the names
Takahashi's transformation, Schrieffer-Wolff transformation, des Cloiseaux
effective Hamiltonian, canonical van Vleck effective Hamiltonian or two-block
orthogonalization method. The effective cluster-additive Hamiltonian and the
transformation for a given subspace of $\mathcal{H}$, that is adiabatically
connected to the eigenspace of $\mathcal{H}_0$ with eigenvalue $e_0^n$, solely
depends on the eigenspaces of $\mathcal{H}$ connected to $e_0^m$ with
$e_0^m\leq e_0^n$. In contrast, other cluster-additive transformations like the
multi-block orthognalization method or perturbative continuous unitary
transformations need a larger basis. This can be exploited to implement the
transformation efficiently both perturbatively and non-perturbatively. As a
benchmark, we perform perturbative and non-perturbative linked-cluster
expansions in the low-field ordered phase of the transverse-field Ising model
on the square lattice for single spin-flips and two spin-flip bound-states.
