  We define quasi-local conserved quantities in general relativity by using the
optimal isometric embedding in [26] to transplant Killing fields in the
Minkowski spacetime back to the 2-surface of interest in a physical spacetime.
To each optimal isometric embedding, a dual element of the Lie algebra of the
Lorentz group is assigned. Quasi-local angular momentum and quasi-local center
of mass correspond to pairing this element with rotation Killing fields and
boost Killing fields, respectively. They obey classical transformation laws
under the action of the Poincar\'e group. We further justify these definitions
by considering their limits as the total angular momentum and the total center
of mass of an isolated system. These expressions were derived from the
Hamilton-Jacobi analysis of gravitation action and thus satisfy conservation
laws. As a result, we obtained an invariant total angular momentum theorem in
the Kerr spacetime. For a vacuum asymptotically flat initial data set of order
$1$, it is shown that the limits are always finite without any extra
assumptions. We also study these total conserved quantities on a family of
asymptotically flat initial data sets evolving by the vacuum Einstein evolution
equation. It is shown that the total angular momentum is conserved under the
evolution. For the total center of mass, the classical dynamical formula
relating the center of mass, energy, and linear momentum is recovered, in the
nonlinear context of initial data sets evolving by the vacuum Einstein
evolution equation. The definition of quasi-local angular momentum provides an
answer to the second problem in classical general relativity on Penrose's list
[18].
