A quantum-classical bracket that satisfies the Jacobi identity

A quantum-classical bracket is proposed and is shown to satisfy the Jacobi identity, in contrast to previous definitions that obey this property only up to higher order terms in the Planck constant $ℏ$⁠. The Jacobi identity is required of a true Lie bracket and ensures that the Lie bracket of constants of motion is also a constant of motion. An explicit calculation of the Jacobi identity highlights the difference between the proposed and traditional definitions. A further example illustrates that the proposed bracket generates a more consistent quantum-classical dynamics than the traditional bracket. The traditional quantum-classical dynamics in the Henon-Heiles system diverges due to higher order $ℏ$ terms. The divergence is eliminated with the proposed bracket.