= Chiral decomposition of the complexified so4 Lie algebra
{c}
{title2=$\mathfrak{so}_4(\mathbb C)\cong\mathfrak{sl}_2(\mathbb C)\oplus\mathfrak{sl}_2(\mathbb C)$}
The self-dual and anti-self-dual combinations of the skew generators form two commuting copies of $\mathfrak{sl}_2(\mathbb C)$. Consequently
$$
\mathfrak{so}_4(\mathbb C)\cong\mathfrak{sl}_2(\mathbb C)\oplus\mathfrak{sl}_2(\mathbb C),
$$
and its <Adjoint representation> decomposes as $(\mathbf3,\mathbf1)\oplus(\mathbf1,\mathbf3)$.
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