- Chi : G -> R^n. This is the coordinate function on the group manifold.
- Chi^-1 : R^n -> G. The inverse of Chi.
- e \in G. The identity element.
- inverse: G -> G. The inverse function (on the group manifold: g -> g^-1).
- *: GxG -> G. The group multiplication function.
See lie-group-SO3.ss for an example of how this is done. It's a bit tricky, because the <chart> (which contains Chi and Chi^-1) must take and produce objects of the <lie-group-element> class, which require the <lie-group> class for one of the slots; but the <lie-group> class requires the <chart> object, so they must be recursively defined.