Metric structures making the cohomology into a module over a Lie algebra

Question

The cohomology of a closed Kaehler manifold is an $\mathfrak{sl}_2$-module. I think Verbitsky has shown that the cohomology of a closed hyperkaehler manifold is an $\mathfrak{so}_5$-module. For what other Lie algebras $\mathfrak{g}$ there exists some metric structure that makes the cohomology of a closed smooth manifold into $\mathfrak{g}$-module? The question is somewhat imprecise but I think the idea is clear.