The finite dimensional representations of the Lorentz group are given by two half-integers $(j_1, j_2)$. One can break up the Lorentz Lie algebra into two $\mathfrak{so}(3)$ Lie algebras, and the half-integers specify the spin of those $\mathfrak{so}(3)$ Lie algebras.
One should be able to do this for the Poincare algebra as well, but for the life of me I can't find a reference or figure out what it is called. I am looking for the discrete representations of the Poincare group, whose sub-representations when restricted to the Lorentz group are the finite dimensional $(j_1, j_2)$ representations.
Note that this is no way shape or form given by usual Wigner classification of unitary irreps of the Poincare group. These representations will not be unitary and that is okay.
