Let $(V, \omega)$ be a symplectic vector space of dimension 6, with a compatible metric $g$. Let $\varphi$ be a 3-form on $V$. Then $\phi$ is primitive, meaning $\star \varphi \wedge \omega = 0$, if and only if $\star \varphi$ is (meaning $\varphi \wedge \omega = 0$).
One possible proof of this could be the following: primitive forms are the unique 12-dimensional irreducible components of $SU(3)$ acting on 3-forms, and $\star$ is $SU(3)$ invariant, hence it must map primitive forms into primitive forms by Schur lemma.
However, I was hoping to have a completely algebraically proof, which I cannot manage to produce because I do not know how to deal with the interaction between the Hodge star and the exterior product. Plus, my proof is very specific to this particular case, and I was hoping an algebraic proof to be more general.