The Koszul homomorphism for a pair of Lie algebras in the theory of exotic characteristic classes of Lie algebroids

•Functoriality and homotopy properties of the exotic characteristic homomorphism for Lie algebroids.•The Koszul homomorphism versus the universal exotic characteristic homomorphism.•Monomorphicity of the universal exotic characteristic homomorphism for a pair of integrable Lie algebroids.•Monomorphicity of the universal exotic characteristic homomorphism for non-integrable Lie algebroids of TC-foliations.

We examine fundamental properties of the universal exotic characteristic homomorphism in the category of Lie algebroids, introduced by the authors in Balcerzak and Kubarski (2012) [4]. The properties under study include: (a) functorial properties with respect to arbitrary morphisms of Lie algebroids, (b) homotopy properties, (c) relationships with the Koszul homomorphism for a pair of isotropy Lie algebras, (d) conditions under which the universal exotic characteristic homomorphism is a monomorphism.