The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie al- gebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical (ρ, η)-system with respect to a (ρ, η)-spray, the Berwald (ρ, η)-derivative operator, and its mixed curva- ture, we obtain main results to conceptualize the Weyl’s method in this general frame- work. Finally, we obtain two new results of Weyl type for the geometry of mechan- ical (ρ, η)-systems. In this way, it is proved that the projectively related sprays first have the same geodesics rather to an increasing parameter transformation and second their Berwald derivatives verify a respective relation.
