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The Coleman–Mandula theorem says that, under its assumptions on a nontrivial analytic relativistic S-matrix and the particle spectrum, every continuous bosonic symmetry algebra is a direct sum of the Poincare algebra and an internal symmetry algebra. Supersymmetry becomes possible by weakening the assumption that the symmetry algebra is an ordinary Lie algebra: a Z2-graded Lie superalgebra admits odd generators whose bracket is an anticommutator. The Haag–Łopuszański–Sohnius theorem then classifies the allowed extension and leads to the Super-Poincaré algebra.

An internal generator $T_a$ is a Lorentz scalar. The Coleman–Mandula theorem and the graded extension allow it to act nontrivially on supercharges only as an R-symmetry. Since R-symmetries are excluded,

Translation invariance of a conserved global supercharge and the graded Jacobi identities giveLorentz covariance requires the supercharges to transform as Weyl spinors,with the complex-conjugate dotted-spinor relation for ${\bar{Q}}_{\dot{\alpha }}$, up to the sign convention used for the action of generators.

The anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}$ transforms as a Lorentz vector because $(1/2,0)\otimes (0,1/2)=(1/2,1/2)$. The only translation generator with that transformation law is $P_{\mu }$, and a normalization of $Q$ fixesAn equal-chirality anticommutator could only contain an antisymmetric spinor contraction times a central charge, but the anticommutator is symmetric under exchange of the complete supercharges. For one supercharge and no central extension this forcesTogether with the stated Poincare brackets, these are the four-dimensional $\mathcal{N}=1$ Super-Poincare relations.