Chiral decomposition of the complexified so4 Lie algebra (${\mathfrak{so}}_4(\mathbb{C})\cong {\mathfrak{sl}}_2(\mathbb{C})\oplus {\mathfrak{sl}}_2(\mathbb{C})$)

The self-dual and anti-self-dual combinations of the skew generators form two commuting copies of ${\mathfrak{sl}}_2(\mathbb{C})$. Consequentlyand its Adjoint representation decomposes as $(3,1)\oplus (1,3)$.