I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie algebras which are defined on field with characteristic zero. In this connection, I infinitesimally deform one of commutation relations (Lie brackets) by adding some terms in its RHS and reach to an infinite dimensional rigid Lie algebra (Witt Lie algebra).The question is:
 Is this rigid Lie algebra unique and independent of how I deform the initial Lie algebra? In fact, if I started with other commutation relations would I reach to another rigid Lie algebra? Is there any theorem about "uniqueness" of rigid Lie algebra which is derived in deformation procedure?