  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 an
extension of this anchored bundle to an infinite rank universal free Lie
algebroid $FR(E)\supset E$. We adapt his construction to the case of an
anchored bundle equipped with an arbitrary connection, $(E,\nabla)$, and show
that it gives rise to a unique connection $\tilde \nabla$ on $FR(E)$ which is
compatible 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 a
Cartan-Lie Algebroid $(A,\bar \nabla)$ factors through a unique Cartan-Lie
algebroid morphism from $(FR(E), \tilde \nabla)$ to $(A,\bar \nabla)$.
  Suppose that, in addition, $M$ is equipped with a geometrical structure
defined by some tensor field $t$ which is compatible with $(E,\rho,\nabla)$ 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 $(E,\rho)$, this condition implies that $M$ carries
a Riemannian foliation. %In general, the compatibility of a tensor $t$ with
$(E,\rho,\nabla)$ implies its adequate invariance transversal to $\rho(E)$. 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 $(FR(E), \tilde \nabla)$.

Differential Geometry