Generalized Bell inequality and a method for its verification

We use the reduced density matrix of the two-particle spin state to construct a generalized Bell–Clauser–Horne–Shimony–Holt inequality. For each specific state and under a special choice of the vectors $\vec a$, $\vec b$, $\vec c$, and $\vec d$, this inequality becomes an exact equality. We show how such vectors can be found using the reduced density matrix. Both sides of this equality have a specific numerical value. We indicate the connection of this number with the measure of entanglement of the two-particle spin state.