  It is well known that the support of an optimal decomposable entanglement
witness is completely entangled. We add two more necessary conditions for the
optimality: The orthogonal complement of the support must have a nonzero
product vector; another one will be given in terms of related faces of a convex
cone. With these necessary conditions, we show that there exist examples of
non-optimal decomposable entanglement witnesses which are the partial
transposes of positive semi-definite matrices supported on completely entangled
spaces, whenever both of the local dimensions are greater than or equal to
three.
