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The coefficients of the formal series must be smooth functions $H_n\in C^{\infty }(M)$, rather than points of $M$. Interpret complex parameter values in the complexified function algebra. The Casimir function condition for the Poisson pencil is a coefficientwise formal identity:Equality of the formal coefficients provesThese are the Lenard-Magri recursion relations, valid for every smooth function $f$.

For $i\ge 1$ and $j\ge 0$, use the recursion in the first argument, then antisymmetry and the recursion in the second:Each step lowers the first index while raising the second. Iterating $i$ times givesbecause $H_0$ is a Casimir function of the first Poisson structure. This includes $i=0$ directly. The recursion one step higher then givesThus the commuting coefficients of a Poisson pencil Casimir satisfyThey are therefore Poisson-commuting functions for every member of the Poisson pencil. The argument proves involution, not functional independence; an infinite sequence can contain repetitions or zero coefficients.