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 which its commutation relations are:
In this connection, I infinitesimally deform the last commutator(the ideal of algebra) by adding some terms in its RHS and reach to an infinite dimensional rigid Lie algebra.The first commutator is a subalgebra which is known as Witt algebra and will remain rigid by deformation procedure. But I am not sure about the second commutator which can be deformed or not. This problem has connection with this question: Is the rigid Lie algebra I mentioned above "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?