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In this talk I want to outline a construction of star products build upon quantizations of symmetries of the manifold. To be more concrete, a normalized 2-cocycle on the universal enveloping algebra of a Lie algebra induces a star product on every manifold the Lie algebra acts on. The approach is functorial and extends to any tensor field of the manifold. In particular, we obtain quantizations of vector fields and differential forms, leading to a twisted Cartan calculus. I give explicit examples of this Drinfel'd twist approach, but I also present symplectic manifolds that can never be quantized in this way. The project is part of a collaboration with my former Master's and PhD supervisors D'Andrea, Esposito, Fiore and Waldmann.