The Poincare and conformal groups are contenders for the most fundamental spacetime symmetry group. An 8-dimensional rep, putting two 4-spinors together, makes a suitable platform to install matrix representations of these two fundamental groups. But some of their generators do not commute, so new generators are introduced to keep the algebra closed. The combined algebra then has 37 basis generators, a dozen more than needed for the Poincare and conformal algebras. Interestingly, with two Lorentz subalgebras, one finds two distinct definitions of spin. For the adjoint representation, one set of Lorentz generators reduces to irreducible representations, all with integer spin. The other Lorentz group reduces to both integer and `half-integer' spin irreducible representations. Also, one finds that the various representations confirm the spin rules for matrix translation generators with the spins of both Lorentz subgroups.