  We continue analysis of \cite{Parsa:2018kys} and study rigidity and stability
of the BMS4 algebra and its centrally extended version. We construct and
classify the family of algebras which appear as deformations of BMS4 and in
general find the four-parameter family of algebras W(a,b;\bar{a},\bar{b}) as a
result of the stabilization analysis, where BMS4=W(-1/2,-1/2;-1/2,-1/2). We
then study the W(a,b;\bar{a},\bar{b}) algebra, its maximal finite subgroups and
stability for different values of the four parameters. We prove stability of
the W(a,b;\bar{a},\bar{b}) family of algebras for generic values of the
parameters. For special cases of (a,b)=(\bar{a},\bar{b})=(0,0) and
(a,b)=(0,-1), (\bar{a},\bar{b})=(0,0) the algebra can be deformed. In
particular we show that centrally extended W(0,-1;0,0) algebra can be deformed
to an algebra which has three copies of Virasoro as a subalgebra. We briefly
discuss these deformed algebras as asymptotic symmetry algebras and the
physical meaning of the stabilization and implications of our result.
