Note: Please see pdf for full abstract with equations.
We consider the primitive decomposition of ∂,∂, Bott-Chern and Aeppli-harmonic (k, k)-forms on compact almost Kähler manifolds (M, J,ω). For any D ∈ {∂, ∂,BC,A}, it is known that the LkP0,0 component of ψ ∈ HDk,k is a constant multiple of ωk up to real dimension 6. In this paper we generalise this result to every dimension. We also deduce information on the components Lk−1P1,1 and Lk−2P2,2 of the primitive decomposion. Focusing on dimension 8, we give a full description of the spaces HBC2,2 and HA2,2, from which follows HBC2,2 ⊆ H∂2,2 and HA2,2 ⊆ H∂2,2. We also provide an almost Kähler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form ψ ∈ HDk,k are not D-harmonic, showing that the primitive decomposition of (k, k)-forms in general does not descend to harmonic forms.