= Three-dimensional Lie algebra structure decomposition
{title2=$C_a{}^b{}_c\epsilon^{acd}=n^{bd}+\epsilon^{bde}a_e$}
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 $C_a{}^b{}_c=\frac12\epsilon_{acd}n^{bd}+\frac12(\delta_c^b a_a-\delta_a^b a_c)$. The only independent triple in the <Jacobi identity> has coefficient $-\frac12n^{be}a_e$, so Jacobi holds exactly when $na=0$. This decomposition is useful in classifying three-dimensional <Lie algebras>.
Back to article page
