= Solution
A <supercharge> is an odd <Weyl spinor> generator. The mixed <anticommutator> has one undotted and one dotted index, hence transforms as a <Lorentz four-vector>. In the minimal <Super-Poincaré algebra>, its only spacetime generator is the <four-momentum>, so covariance fixes it to $c\sigma^\mu_{\alpha\dot\beta}P_\mu$. Normalizing the charges canonically fixes $c=2$:
$$
\boxed{\{Q_\alpha,\bar Q_{\dot\beta}\}
=2\sigma^\mu_{\alpha\dot\beta}P_\mu.}
$$
The number two is a normalization, not something Lorentz covariance alone could fix under $Q\mapsto aQ$.
For an explicit closure calculation, use left <Grassmann derivatives> in <superspace> and the differential translation convention $P_\mu=i\partial_\mu$:
$$
Q_\alpha=\frac\partial{\partial\theta^\alpha}
-i\sigma^\mu_{\alpha\dot\beta}\bar\theta^{\dot\beta}\partial_\mu,
\qquad
\bar Q_{\dot\beta}=-\frac\partial{\partial\bar\theta^{\dot\beta}}
+i\theta^\alpha\sigma^\mu_{\alpha\dot\beta}\partial_\mu.
$$
Each of the two derivative-coordinate cross terms contributes $i\sigma^\mu\partial_\mu$; the other terms cancel by anticommutativity. This proves the displayed coefficient. The same calculation gives $\{Q,Q\}=\{\bar Q,\bar Q\}=0$ and $[P_\mu,Q]=[P_\mu,\bar Q]=0$. There is no scalar central charge for a single <supersymmetry> label: symmetry of the <anticommutator> is incompatible with a term proportional only to the antisymmetric $\epsilon_{\alpha\beta}$.
In the massive rest frame $P_\mu=(m,0,0,0)$, the mixed relation is $\{Q_\alpha,Q_\beta^\dagger\}=2m\delta_{\alpha\beta}$. These positive <canonical anticommutation relations> supply the input for the normalization and finite <fermionic Fock space> in the question-level continuation.
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