Sorting out minus signs in differential geometry

in the Lie algebra of vector fields on a smooth manifold

Bug

The usual definition of commutator as derivations of and infinitesimal generators of Lie group actions

produces an anti-homomorphism . This is expected because acts on on the right.

• settle with the brackets
  • does it agree with brackets in ?
  • settle with the anti-homomorphism
  • use as definition of infinitesimal generators
    • this seems to produce as the infinitesimal generator of flow of
• use the brackets , possibly defined using flows/adjoint action

in the Hamiltonian vector field of a smooth function on a symplectic manifold

Hamiltonian vector field of is

where is the symplectic form.

Why the minus sign?