  We give a simple and constructive extension of Rai\c{t}\u{a}'s result that
every constant-rank operator possesses an exact potential and an exact
annihilator. Our construction is completely self-contained and provides an
improvement on the order of the operators constructed by Rai\c{t}\u{a}, as well
as the order of the explicit annihilators for elliptic operators due to Van
Schaftingen. We also give an abstract construction of an optimal annihilator
for constant-rank operators, which extends the optimal construction of Van
Schaftingen for elliptic operators. Lastly, we establish a generalized
Poincar\'e lemma for constant-rank operators and homogeneous spaces on
$\mathbb{R}^d$, and we prove that the existence of potentials on spaces of
periodic maps requires a strictly weaker condition than the constant-rank
property.
