Given the following Christoffel symbols for a metric on $\mathbb{H}^3$, how do I obtain the structure constants $C^i_{jk}$, such that $[\theta_i,\theta_k] = C^k_{ij}\theta_k$, and get the to agree with what I have? :
$$\Gamma^3_{11} = z^{-1}, \Gamma^1_{13}=-z^{-1}, \Gamma^3_{22}=z^{-1}$$ $$\Gamma^2_{23} = -z^{-1}, \Gamma^1_{31} = -z^{-1}, \Gamma^2_{32} = -z^{-1} $$ $$\Gamma^3_{33} = -z^{-1}$$
I have seen formulae such as $\Gamma^i_{jk} =\frac{1}{2}(C^i_{jk}+C^k_{ij} + C^k_{ji}) $, but I have already calculated these $C^i_{jk}$ using Bianchi classifications and they are just a bunch of $1$ and $-1$.