= Real four-dimensional rotation algebra splitting
{title2=$\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$}
For antisymmetric Maurer–Cartan forms with $d\lambda_{ij}=\lambda_{ik}\wedge\lambda_{kj}$, let $\alpha^\pm=(\lambda_{12}\pm\lambda_{43},\lambda_{23}\pm\lambda_{41},\lambda_{31}\pm\lambda_{42})$. Direct expansion gives $d\alpha_1^\pm=-\alpha_2^\pm\wedge\alpha_3^\pm$ and cyclic analogues, with no mixed terms. The dual triples form two commuting ideals, each with the <SO(3) Lie algebra> brackets. This is a real compact splitting, distinct from the splitting of the complexified algebra into two copies of $\mathfrak{sl}_2(\mathbb C)$.
Back to article page
