Simplicity of Lie algebras of Poisson brackets

Abstract: Let $A$ be an associative commutative algebra with $1$ over a field of zero characteristic, ${,} : A \times A \to A$ is a Poisson bracket, $Z = { a \in A \mid {a, A} = (0) }.$ We prove that if $A$ is simple as a Poisson algebra then the Lie algebra $\frac{{A,A}}{{A,A}\cap Z}$ is simple.