I'm reading a book called "Physics from symmetry" by Schwichtenberg. The author talks about generators $\{J_i\}$ of the Lie algebra of $SO(3)$ and he derives the conditions that they have to fulfill:
$$ J^T + J = 0 $$ $$ \mathrm{tr} J = 0 $$
He finds the following basis:
$$ J_1 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} $$
$$ J_2 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{bmatrix} $$
$$ J_3 = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$
It is possible to write, as the author say, that:
$$ (J_i)_{jk} = -\varepsilon_{ijk} $$
Then the author claims that by computing the Lie braket, using the found generators, it is possible to write that:
$$ [J_i, J_j] = J_i J_j - J_j J_i = \varepsilon_{ijk} J_k $$
This is the statement I want to prove and so far I came up with this line of reasoning: $(J_i)_{jk} = J_{ijk}$ is a tensor since the Levi-Civita symbol, which $J_i$ can be written in terms of, is itself a tensor. Therefore it is possible to write:
$$ [J_i, J_j]_m^n = (J_i)_{mk} (J_j)^{kn} - (J_j)_{mk} (J_i)^{kn} = J_{imk} J_j^{\ \ kn} - J_{jmk} J_i^{\ \ kn} $$
lowering the indices of the second generator of the couple:
$$ J_{imk} J_j^{\ \ kn} - J_{jmk} J_i^{\ \ kn} = g^{ks} g^{np} J_{imk} J_{jsp} - g^{ks} g^{np} J_{jmk} J_{isp} $$
Then using the fact that $J_{ijk} = -\varepsilon_{ijk}:$
$$ g^{ks} g^{np} \varepsilon_{imk} \varepsilon_{jsp} - g^{ks} g^{np} \varepsilon_{jmk} \varepsilon_{isp} = \varepsilon_{imk} \varepsilon_j^{\ \ kn} - \varepsilon_{jmk} \varepsilon_i^{\ \ kn} = $$ $$ = g_{jl}\varepsilon_{imk} \varepsilon^{lkn} - g_{il}\varepsilon_{jmk} \varepsilon^{lkn} $$ $$ = - g_{jl}\varepsilon_{imk} \varepsilon^{lnk} + g_{il}\varepsilon_{jmk} \varepsilon^{lnk} $$
from which, knowing that $\varepsilon_{mno} \varepsilon^{pqr} = \delta_{mno}^{pqr}$ and that $\delta_{mnk}^{pqk} = 2\delta_{mn}^{pq}$:
$$ - g_{jl}\varepsilon_{imk} \varepsilon^{lnk} + g_{il}\varepsilon_{jmk} \varepsilon^{lnk} = $$
$$ = - g_{jl} \delta_{imk}^{lnk} + g_{il} \delta_{jmk}^{lnk} = $$ $$ = - 2 g_{jl} \delta_{im}^{ln} + 2 g_{il} \delta_{jm}^{ln} = $$
I'm stuck at this point. I'm not entirely sure that I can treat $J_{ijk}$ as a tensor: the justification I adduced seems pretty weak, as it realies on a way of writing $J_{ijk}$ in terms of $\varepsilon_{ijk}$ and not on the defining property of tensors. Is the reasoning I'm following correct? And in that case, how do I continue?
P.S.: thanks in advance for any eventual answer and excuse my poor english, as I'm still practising it!