Cartan calculus for $C^\infty$-ringed spaces

In an earlier paper (arXiv:2212.11163) I constructed a complex of differential forms on a local $C^\infty$-ringed space. In this paper I define a sheaf of vector fields (``the tangent sheaf'') on a local $C^\infty$-ringed space, define contractions of vector fields and forms, Lie derivatives of forms with respect to vector fields, and show that the standard equations of Cartan calculus hold for vector fields and differential forms on local $C^\infty$-ringed spaces.