  The study of mechanical systems on Lie algebroids permits an understanding of
the dynamics described by a Lagrangian or Hamiltonian function for a wide range
of mechanical systems in a unified framework. Systems defined in tangent
bundles, Lie algebras, principal bundles, reduced systems and constrained are
included in such description.
  In this paper, we investigate how to derive the dynamics associated with a
Lagrangian system defined on the set of admissible elements of a given Lie
algebroid using Tulczyjew's triple on Lie algebroids and constructing a
Lagrangian Lie subalgebroid of a symplectic Lie algebroid, by building on the
geometric formalism for mechanics on Lie algebroids developed by M. de Le\'on,
J.C. Marrero and E. Mart\'inez on "Lagrangian submanifolds and dynamics on Lie
algebroids".
