= Commuting coefficients of a Poisson pencil Casimir
{title2=$\{H_i,H_j\}_1=\{H_i,H_j\}_2=0$}
For a <Lenard-Magri recursion>, antisymmetry shifts an index: $\{H_i,H_j\}_1=\{H_{i-1},H_{j+1}\}_1$ when $i\geq1$. Repeating reaches $\{H_0,H_{i+j}\}_1=0$, because $H_0$ is a <Casimir function>. The recursion then gives $\{H_i,H_j\}_2=-\{H_{i+1},H_j\}_1=0$. This proof applies to every finite initial portion for the indices that can be shifted inside it, and to all pairs in an infinite sequence.
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