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Is there a classification of pairs $(\mathfrak g,V)$, with $\mathfrak g$ a Lie algebra and $V$ a $\mathfrak g$-module, such that $(\Lambda^3V)^{\mathfrak g}\neq0$?

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Not a full answer, but you can get plenty of examples by taking $V$ to be the coadjoint representation of a Lie algebra with nonzero Killing form. Then $\omega \in \Lambda^3\mathfrak{g}^*$, defined by $$\omega(X,Y,Z) = \mathrm{Tr}~\mathrm{ad}([X,Y])\mathrm{ad}(Z),$$ is invariant and nonzero.

Similarly, any metric Lie algebra with $V$ the adjoint representation also works. The invariant 3-form is then $$\omega(X,Y,Z) = \langle [X,Y], Z\rangle,$$ with $\langle-,-\rangle$ the ad-invariant inner product.

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