Metrizability of the Lie algebroid generalized tangent bundle and (generalized) Lagrange $(\rho,\eta)$-spaces

A class of metrizable vector bundles in the general framework of generalized Lie algebroids have been presented in the eight reference. Using a generalized Lie algebroid we obtain the Lie algebroid generalized tangent bundle of a vector bundle. This Lie algebroid is a new example of metrizable vector bundle. A new class of Lagrange spaces, called by use, generalized Lagrange (\rho?;\eta?)-space, Lagrange (\rho?;\eta?)-space and Finsler (\rho?;\eta?)-space are presented. In the particular case of Lie algebroids, new and important results are presented. In particular, if all morphisms are identities morphisms, then the classical results are obtained.