I have been learning about the $SO(3)$ symmetry group in my physics classes, and it is often asserted in books that the structure constants of the Lie algebra behave just as indices do in Euclidean space where one can largely ignore the distinction between vectors and co-vectors. However because the structure constants are basis dependant this seems impossible unless one picks a particular basis. However in the canonical basis most use:
\begin{equation}\begin{array}{ccc} L_{1}=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array}\right) &L_{2}=\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array}\right) &L_{3}=\left(\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) \end{array} \end{equation}
We have the positive definite metric induced from the Killing form: $$ g_{ij}=-\mathrm{Tr}(L_iL_j)=2\delta_{ij} $$ Which is not Euclidean. This means that while the $c$ below are all $1$, $0$, or $-1$ $$ [L_i, L_j]=c_{ij}^{\>\>\>k}L_k $$ the lowered structure constants cannot be merely $c_{ijk} = \varepsilon_{ijk}$ where $\varepsilon$ is the Levi-Civita Tensor
In fact the lowered structure constants would have to be twice the Levi-Civita Tensor.
Have I made an error? Does it have to do with Levi-Civita tensor in fact being a Tensor Density? If not, why do physicists insist it makes no difference, when it clearly changes the constants?