I'm sorry if my question is too classic and basic. As Dirac-Bergmann algorithm for Hamiltonian formalism, I find out that a Hamiltonian system is inconsistent if Poisson bracket of primary constraints and Hamiltonian $$\{f_{i},H\}=1\approx 0$$ If the Poisson brackets are equal to $0$ or produce new constraints, the Hamiltonian system is consistent. Am I right about them? Working on a trivial example, such as: $$H=\frac{1}{2}(p^2+x^2)\\f=p^2+x^2+x^4$$ I take Poisson bracket between them and obtain $4x^3p$. So this is consistent? Because there is no obvious result as $1$ appeared.

I'm sorry if my question is too classic and basic. As Dirac-Bergmann algorithm for Hamiltonian formalism, I find out that a Hamiltonian system is inconsistent if Poisson bracket of primary constraints and Hamiltonian $$\{f_{i},H\}=1\approx 0$$ If the Poisson brackets are equal to $0$ or produce new constraints, the Hamiltonian system is consistent. Am I right about them? Working on a trivial example, such as: $$H=\frac{1}{2}(p^2+x^2)\\f=p^2+x^2+x^4$$ I take Poisson bracket between them and obtain $4x^3p$. So this is consistent? Because there is no obvious result as $1$ appearing.