Real four-dimensional rotation algebra splitting ($\mathfrak{so}(4)\cong \mathfrak{so}(3)\oplus \mathfrak{so}(3)$)

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 ${\mathfrak{sl}}_2(\mathbb{C})$.