Three-dimensional Lie algebra structure decomposition ($C_a{}^b{}_c{ϵ}^{acd}=n^{bd}+{ϵ}^{bde}{a}_e$)

An oriented three-dimensional Lie algebra has its antisymmetric structure constants encoded by the displayed matrix, whose symmetric part is $n^{bd}$ and whose antisymmetric part uniquely defines the covector $a_e$. In a volume-normalized basis, the inverse is $C_a{}^b{}_c=\frac{1}{2}{ϵ}_{acd}{n}^{bd}+\frac{1}{2}({\delta }_c^b{a}_a-{\delta }_a^b{a}_c)$. The only independent triple in the Jacobi identity has coefficient $-\frac{1}{2}{n}^{be}{a}_e$, so Jacobi holds exactly when $na=0$. This decomposition is useful in classifying three-dimensional Lie algebras.