  We define the 2-Toda lattice on every simple Lie algebra g, and we show its
Liouville integrability. We show that this lattice is given by a pair of
Hamiltonian vector fields, associated with a Poisson bracket which results from
an R-matrix of the underlying Lie algebra. We construct a big family of
constants of motion which we use to prove the Liouville integrability of the
system. We achieve the proof of their integrability by using several results on
simple Lie algebras, R-matrices, invariant functions and root systems.
