  One (actually, almost the only effective) way to prove formality of a
differentiable manifold is to be able to produce a suitable derivation $\delta$
such that $d\delta$-lemma holds. We first show that such derivation $\delta$
generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the
supercommutation of $d$ and $\delta$ (which is a natural, essentially necessary
condition to get a $d\delta$-lemma) is equivalent to vanishing of the
Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that
ensure the $d\delta$-lemma holds: we consider the cases when $R$ is self
adjoint with respect to a Riemannian metric or compatible with an almost
symplectic structure. Finally, we show that if $R$ is scew-symmetric with
respect to a Riemannian metric, has constant determinant, and if its Nijenhujis
torsion vanishes, then the orthogonal component of $R$ in its polar
decomposition is a complex structure compatible with the metric, which gives us
a new characterization of K\"ahler structures
