Optimal Reduction by Stages

As we already saw in Part II, the reduction by stages procedure consists of carrying out reduction in two shots using the normal subgroups of the symmetry group. To be more specific, suppose that we are in the same setup as Theorem 13.5.1 and that the symmetry group G has a closed normal subgroup N. In this chapter we will spell out the conditions under which optimal reduction by G renders the same result as reduction in the following two stages: we first reduce by N; the resulting space inherits symmetry properties coming from the quotient Lie group G/N that can be used to reduce one more time.