  The geometry of the Lie algebroid generalized tangent bundle of a generalized
Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and
Bianchi type are presented. Introducing the notion of geodesic of a mechanical
$\left( \rho ,\eta \right) $-system with respect to a $(\rho, \eta)$-spray, the
Berwald $(\rho, \eta)$-derivative operator and its mixed curvature, we obtain
main results to conceptualize the Weyl's method in this general framework.
Finally, we obtain two new results of Weyl type for the geometry of mechanical
$\left( \rho ,\eta \right) $-systems.
