Geometric intuition behind Jacobi Identity

Question

I'm currently trying to understand the Lie bracket of smooth vector fields.The bracket operation satisfies the Jacobian Identity: If X and Y are two smooth vector fields on a manifold, we define their Lie bracket as $[X,Y]=XY-YX.$ Using that definition, we have $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0. $ I am familiar with the Lie derivative and I know the bracket operation coincides with the Lie derivative. I'm wondering if there is a geometric intuition behind this identity.