so4 Lie algebra (${\mathfrak{so}}_4$)

The complexified so4 Lie algebra consists of four-by-four complex skew-symmetric matrices and has dimension six. Its root system is $D_2=A_1\times A_1$, reflecting its decomposition into two simple rank-one ideals.

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 $(3,1)\oplus (1,3)$.