Reduction is a process that uses symmetry to lower the order of a Hamiltonian
system. The new variables in the reduced picture are often not canonical: there
are no clear variables representing positions and momenta, and the Poisson
bracket obtained is not of the canonical type. Specifically, we give two
examples that give rise to brackets of the noncanonical Lie-Poisson form: the
rigid body and the two-dimensional ideal fluid. From these simple cases, we
then use the semidirect product extension of algebras to describe more complex
physical systems. The Casimir invariants in these systems are examined, and
some are shown to be linked to the recovery of information about the
configuration of the system. We discuss a case in which the extension is not a
semidirect product, namely compressible reduced MHD, and find for this case
that the Casimir invariants lend partial information about the configuration of
the system.