rank tensor (the cosymplectic form) defined by the Poisson bracket. This is a natural strategy: many of the Casimir invariants in [@mrmp] were calculated in this fashion, a fact not noted there. This method [*has*]{} been presented explicitly, for infinite-dimensional brackets in [@kuro] and finite-dimensional brackets in [@byv2]. Here, we show that these papers both treat special cases of a single method by presenting the method in a geometrical framework that includes both cases (Section \[S-CAS\]). This presentation has a partly pedagogical function: we present a transparent example of differential geometry applied to Hamiltonian dynamics. But also, it offers a clear framework for arriving at a novel result: the form of Casimir invariants for Poisson brackets with a nested Lie-Poisson structure, involving both finite and infinite-dimensional brackets. The method is used to compute these Casimir invariants (and those of several other Poisson brackets) in Sections \[S-FINITE\] and \[S-INFINITE\].

Casimir invariants and Pfaffian systems {#S-CAS}
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The Poisson bracket is a mathematical structure common to all Hamiltonian systems. If $z^{\alpha}$ denotes a coordinate of a point in a phase space M, the equations of motion generated by a Hamiltonian $H \colon {\rm M}
\rightarrow I \!\! R$ are $\dot z^{\alpha} = 