A note on Chern-Weil classes of Cartan connections

We present a construction of Chern-Weil characteristic classes for pairs Cartan geometries sharing the same underlying data. Given any model Cartan geometry $(Q,ω)$ with underlying data $(G,V)$ and a second Cartan geometry $(P,θ)$ with the same underlying data, we define a subalgebra of polynomials on the Atiyah algebroid of $Q$ together with a characteristic map that recovers the classical Chern-Weil map of a Cartan connection when $(Q,ω)$ arises from a Klein pair modeling $(P,θ)$.