= Solution
In either the displayed <Dirac representation of the gamma matrices> or the displayed <Weyl representation of the gamma matrices>, the spatial blocks give
$$
[\gamma^j,\gamma^k]=-\begin{pmatrix}[\sigma^j,\sigma^k]&0\\0&[\sigma^j,\sigma^k]\end{pmatrix}.
$$
The <Pauli matrix commutator identity> $[\sigma^j,\sigma^k]=2i\epsilon^{jkl}\sigma^l$ therefore yields
$$
\boxed{S^{jk}=-\frac i2\epsilon^{jkl}\begin{pmatrix}\sigma^l&0\\0&\sigma^l\end{pmatrix}.}
$$
The <Levi-Civita symbol> is normalized by $\epsilon^{123}=1$. The constant is $-i/2$, consistent with rotation generators represented by <skew-Hermitian matrices> under the chosen <Lorentz algebra> convention. For $j=k$ both sides vanish automatically.
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