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Consider an anchored bundle $(E,\rho)$, i.e. a vector bundle $E\to M$equipped with a bundle map $\rho \colon E \to TM$ covering the identity.M.~Kapranov showed in the context of Lie-Rinehard algebras that there exists anextension of this anchored bundle to an infinite rank universal free Liealgebroid $FR(E)\supset E$. We adapt his construction to the case of ananchored bundle equipped with an arbitrary connection, $(E,\nabla)$, and showthat it gives rise to a unique connection $\tilde \nabla$ on $FR(E)$ which iscompatible with its Lie algebroid structure, thus turning $(FR(E), \tilde\nabla)$ into a Cartan-Lie algebroid. Moreover, this construction is universal:any connection-preserving vector bundle morphism from $(E,\nabla)$ to aCartan-Lie Algebroid $(A,\bar \nabla)$ factors through a unique Cartan-Liealgebroid morphism from $(FR(E), \tilde \nabla)$ to $(A,\bar \nabla)$.  Suppose that, in addition, $M$ is equipped with a geometrical structuredefined by some tensor field $t$ which is compatible with $(E,\rho,\nabla)$ inthe sense of being annihilated by a natural $E$-connection that one canassociate to these data. For example, for a Riemannian base $(M,g)$ of aninvolutive anchored bundle $(E,\rho)$, this condition implies that $M$ carriesa Riemannian foliation. %In general, the compatibility of a tensor $t$ with$(E,\rho,\nabla)$ implies its adequate invariance transversal to $\rho(E)$. Itis shown that every $E$-compatible tensor field $t$ becomes invariant withrespect to the Lie algebroid representation associated canonically to theCartan-Lie algebroid $(FR(E), \tilde \nabla)$.

Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries