Nilpotent Lie algebras of breadth type (0,3)

For a natural number m, a Lie algebra L over a field k is said to be of breadth type (0,m) if the co-dimension of the centralizer of every non-central element is m. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type (0,3) over F-q of odd characteristic up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, C and R. We also discuss 2-step nilpotent Camina Lie algebras.