  In this article, a unified approach to obtain symplectic integrators on T*G
from Lie group integrators on a Lie group G is presented. The approach is
worked out in detail for symplectic integrators based on
Runge--Kutta--Munthe-Kaas methods and Crouch--Grossman methods. These methods
can be interpreted as symplectic partitioned Runge--Kutta methods extended to
the Lie group setting in two different ways. In both cases, we show that it is
possible to obtain symplectic integrators of arbitrarily high order by this
approach.
