Double bracket vector fields on Poisson manifolds

We generalize the double bracket vector fields defined on compact semi-simple Lie algebras to the case of general Poisson manifolds endowed with a pseudo-Riemannian metric. We construct a generalization of the normal metric such that the above vector fields, when restricted to a symplectic leaf, become gradient vector fields. We illustrate the discussion at a variety of examples and carefully discuss complications that arise when the pseudo-Riemannian metric does not induce a non-degenerate metric on parts of the symplectic leaves.