For an oriented Riemannian 4-manifold, we have the Levi-Civita connection associated natually to the metric. This connection also gives us a linear connection on all tensor fields over the manifold, especially for two forms.
We have w.r.t. this metric, the decomposition of space of two forms into +1 and -1 eignespaces of Hodge star operator.
My question is, how can we prove that the Levi-Civita connection preserves this decomposition? i.e. for any tangent field $X$, if $\omega$ is a self-dual 2-form, do we also have that the $\nabla_X\omega$ is also a self-dual 2-form?
Is there any easier way to see this?