Deforming a Lie algebra by means of a 2-form

We consider a vector space VV over K=RK=R or CC, equipped with a skew symmetric bracket [⋅,⋅]:V×V→V[⋅,⋅]:V×V→V and a 2-form ω:V×V→Kω:V×V→K. A simple change of the Jacobi identity to the form [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=ω(B,C)A+ω(A,B)C+ω(C,A)B[A,[B,C]]+[C,[A,B]]+[B,[C,A]]=ω(B,C)A+ω(A,B)C+ω(C,A)B opens up new possibilities, which shed new light on the Bianchi classification of three-dimensional Lie algebras.