Consider a homogeneous space given by the quotient of two Lie groups $G/H=M$. Suppose that $M$ is endowed with a $G$-invariant metric $g$ ($M$ has a natural left action of $G$).
Then this metric induces a (unique) torsion-free connection $\nabla$ on $M$. Should I find any $\nabla$-parallel differential $k$-form, I would already know that said $k$-form is necessarily closed, thanks to this neat identity: $$ (\text{d} \alpha)(v_1,\dots,v_{k+1}) = \sum_{i=1}^{k+1} (-1)^{i+1} (\nabla_{v_i}\alpha) (v_1,\dots,\hat{v_i},\dots,v_{k+1}) $$ How could I then extend this result to deduce (if true!) that any $G$-invariant $k$-form is necessarily closed? If not true, what would be a counterexample? I'm basing my stratigy in some naive intuition that $G$-acts in "very special ways" in the homogeneous space, that could be completely off.
As an extra: could one modify this construction to show that a $G$-invariant almost complex structure on $J$ is integrable?