  We present a novel construction of linear deformations for Lie algebras and
use it to prove the non-rigidity of several classes of Lie algebras in
different varieties. We consider the family of Lie algebras with an abelian
factor showing that, in general, they are not rigid even for the case of a
1-dimensional abelian factor. We also address the problem of k-rigidity for
k-step nilpotent Lie algebras. Using our construction and recent results in
[BCC] we prove the that the k-step free nilpotent Lie algebras are k-rigid but
not (k + 1)-rigid and that Heisenberg Lie algebras are 2-rigid but not 3-rigid.
