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

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),∇̃).