  For any compact almost complex manifold $(M,J)$, the last two authors defined
two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology
group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology
classes which can be represented by $J$-invariant, respectively,
$J$-anti-invariant real $2-$forms. In this note, it is shown that in dimension
4 these subgroups induce a cohomology decomposition of $H^2(M, \mathbb{R})$.
This is a specifically 4-dimensional result, as it follows from a recent work
of Fino and Tomassini. Some estimates for the dimensions of these groups are
also established when the almost complex structure is tamed by a symplectic
form and an equivalent formulation for a question of Donaldson is given.
