We consider a skew-symmetric n-ary bracket on the polynomial algebra K[x1,…,xn,xn+1] (n≥2) over a field K of characteristic zero defined by {a1,…,an}=Jac(a1,…,an,C), where C is a fixed element of K[x1,…,xn,xn+1] and Jac is the Jacobian. If n=2 then this bracket is a Poisson bracket and if n≥3 then it is an n-Lie-Poisson bracket on K[x1,…,xn,xn+1]. We describe the center of the corresponding n-Lie-Poisson algebra and show that the quotient algebra K[x1,…,xn,xn+1]/(C−λ), where (C−λ) is the ideal generated by C−λ, 0≠λ∈K, is a simple central n-Lie-Poisson algebra if C is a homogeneous polynomial that is not a proper power of any nonzero polynomial. This construction includes the quotients P(sl2(K))/(C−λ) of the Poisson enveloping algebra P(sl2(K)) of the simple Lie algebra sl2(K), where C is the standard Casimir element of sl2(K) in P(sl2(K)). It is also proven that the quotients P(M)/(CM−λ) of the Poisson enveloping algebra P(M) of the exceptional simple seven dimensional Malcev algebra M are central simple, where CM is the standard Casimir element of M in P(M).