= Solution
Choose $P_\mu=-i\partial_\mu$ and left Grassmann derivatives. A consistent differential realization of <four-dimensional N=1 supersymmetry> is
$$
\mathcal Q_\alpha=i\frac{\partial}{\partial\theta^\alpha}-(\sigma^\mu\bar\theta)_\alpha\partial_\mu,
\qquad
\overline{\mathcal Q}_{\dot\alpha}=-i\frac{\partial}{\partial\bar\theta^{\dot\alpha}}+(\theta\sigma^\mu)_{\dot\alpha}\partial_\mu.
$$
The signs can change with the definitions of $P_\mu$, Grassmann differentiation and the finite transformation, but these operators obey the convention-independent content of the <Super-Poincaré algebra>,
$$
\{\mathcal Q_\alpha,\overline{\mathcal Q}_{\dot\beta}\}=2\sigma^\mu_{\alpha\dot\beta}P_\mu,
\qquad
\{\mathcal Q_\alpha,\mathcal Q_\beta\}
=\{\overline{\mathcal Q}_{\dot\alpha},\overline{\mathcal Q}_{\dot\beta}\}=0.
$$
They follow by expanding the finite transformation to first order and requiring a supersymmetry translation of the <superspace> coordinates.
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