= Solution
Use mostly-minus <metric signature>, $\eta=\operatorname{diag}(1,-1,-1,-1)$. The <Weyl sigma matrices> are $\sigma^\mu=(I,\boldsymbol\sigma)$ and $\bar\sigma^\mu=(I,-\boldsymbol\sigma)$, with the three <Pauli matrices>
$$
\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad \sigma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad \sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
$$
The <Pauli matrix multiplication law> $\sigma_i\sigma_j=\delta_{ij}I+i\epsilon_{ijk}\sigma_k$ immediately gives $\sigma^\mu\bar\sigma^\nu+\sigma^\nu\bar\sigma^\mu=2\eta^{\mu\nu}I$: the $00$ case is $2I$, the mixed cases cancel, and the spatial cases give $-2\delta_{ij}I$. The spinor indices of $\sigma^\mu_{\alpha\dot\beta}$ express the correspondence between a <Lorentz four-vector> and a Hermitian $2\times2$ matrix. For $X=x^0I+x^i\sigma_i$, $\det X=(x^0)^2-|\mathbf x|^2$; $X\mapsto MXM^\dagger$ preserves this <determinant> for $M\in\mathrm{SL}(2,\mathbb C)$, yielding the <Lorentz spinor double cover>.
In <four-dimensional N=1 supersymmetry>, the odd <supercharges> $Q_\alpha$ and $\bar Q_{\dot\alpha}=Q_\alpha^\dagger$ transform as conjugate <Weyl spinors>. Their <Super-Poincaré algebra> is
$$
\boxed{\{Q_\alpha,\bar Q_{\dot\beta}\}=2\sigma^\mu_{\alpha\dot\beta}P_\mu,\qquad \{Q_\alpha,Q_\beta\}=\{\bar Q_{\dot\alpha},\bar Q_{\dot\beta}\}=0,\qquad [P_\mu,Q_\alpha]=[P_\mu,\bar Q_{\dot\alpha}]=0.}
$$
The ordinary translation and Lorentz generators satisfy the <Poincare algebra>, and the Lorentz generators act on the <supercharges> by the corresponding <group representations> on <Weyl spinors>. Tracing the mixed <anticommutator> gives $P_0=\frac14\sum_\alpha\{Q_\alpha,Q_\alpha^\dagger\}$, proving <energy positivity in global supersymmetry>. For a massive rest-frame state, $P_\mu=(M,0,0,0)$, the rescaled charges $Q_\alpha/\sqrt{2M}$ obey the <canonical anticommutation relations>. Acting with their adjoints builds a <supermultiplet> containing states of alternating <fermion parity>.
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