Consider an anchored bundle (E,ρ), i.e. a vector bundle E→M equipped with a bundle map ρ:E→TM covering the identity. M. Kapranov showed in the context of Lie–Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid FR(E)⊃E. We adapt his construction to the case of an anchored bundle equipped with an arbitrary connection, (E,∇), and show that it gives rise to a unique connection ∇̃ on FR(E) which is compatible with its Lie algebroid structure, thus turning (FR(E),∇̃) into a Cartan–Lie algebroid. Moreover, this construction is universal: any connection-preserving vector bundle morphism from (E,∇) to a Cartan–Lie Algebroid (A,∇̄) factors through a unique Cartan–Lie algebroid morphism from (FR(E),∇̃) to (A,∇̄).Suppose that, in addition, M is equipped with a geometrical structure defined by some tensor field t which is compatible with (E,ρ,∇) in the sense of being annihilated by a natural E-connection that one can associate to these data. For example, for a Riemannian base (M,g) of an involutive anchored bundle (E,ρ), this condition implies that M carries a Riemannian foliation. It is shown that every E-compatible tensor field t becomes invariant with respect to the Lie algebroid representation associated canonically to the Cartan–Lie algebroid (FR(E),∇̃).