It is widely asserted that the Jacobi identity for the Poisson bracket, { f , { g , h } } + { g , { h , f } } + { h , { f , g } } = 0 follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is sufficient to show that: ad { g , f } = ad − − { f , g } = [ ad f , ad g ] where the operator ad g on smooth functions on M is defined by ad g ⁡ ⁡ ( ⋅ ⋅ ) = { ⋅ ⋅ , g } and the bracket on the right-hand side is the commutator of operators, [ A , B ] = A ⁡ ⁡ B − − B ⁡ ⁡ A . By (1), the operator ad g is equal to the operator X g. The proof of the Jacobi identity follows from (3) because, up to the factor of -1, the Lie bracket of vector fields is just their commutator as differential operators.