Poincaré group and BMS symmetry How we can derive BMS symmetry by Poincaré group?
 A: There's no way to derive the BMS group from the Poincaré group, because the BMS group is truly a generalization of the Poincaré group.  For example, the BMS group is not a symmetry of the Minkowski spacetime, which is essentially the defining feature of the Poincaré group.(*)  It is true that the Poincaré group is a subgroup of the BMS group, but there is strictly more information required to understand the BMS group — specifically the asymptotic nature of the BMS symmetry.
However, there are ways to "derive" both groups.  And the methods are pretty systematic, so the process is similar for both groups.  Generally, you start by looking for the Lie algebras (the infinitesimal behavior of the groups themselves).  Via the exponential map, this gets you part of each group — the part that is continuously connected to the identity.  Then you have discrete symmetries like parity inversion and time reversal that arise from other considerations.
In both cases, you start off with the metric and look for infinitesimal coordinate transformations such that the transformed metric is the same function of the new coordinates as the original metric was of the original coordinates.  These infinitesimal coordinate transformations are assumed to be differentiable, so they result in a vector field $\xi^a$.  We know the effect any infinitesimal coordinate transformation has on any metric: it changes the metric as
\begin{equation}
  \delta g_{ab} = - \nabla_b\xi_{a} - \nabla_a\xi_{b}.
\end{equation}
If this transformation is a symmetry of the metric, then $\xi^a$ must be a Killing vector field.  It's not too hard to look for all the possible solutions of Killing's equation
\begin{equation}
  \nabla_b\xi_{a} + \nabla_a\xi_{b} = 0.
\end{equation}
In the case of the Minkowski spacetime, I just googled very briefly and found a nice derivation of the Killing vector fields in section 3 of these notes.  For BMS, you need to look at the asymptotic behavior of the metric to figure out the Killing vector fields.  This is exactly what Sachs does in section III of his original paper explaining the BMS group.  In each case, the Killing vector fields are precisely the generators that make up the Lie algebra.

(*) The Erlangen program reverses this logic, by defining Minkowski spacetime as a homogeneous space of the Poincaré group, but I think it's fair to say that it's more common to start from Minkowski spacetime and derive the Poincaré group as its group of symmetries.
