Vogel assigns to every simple metric Lie algebra (and more generally to every simple metric Lie algebra object in a symmetric monoidal category) a point in the orbifold $\mathbb{P}^2/S_3$ (where $S_3$ acts by permuting the 3 projective coordinates) based on the value of the Casimir on the various summands of the symmetric square of the adjoint representation. These three numbers are only defined up to permutation (as there's no natural way to specify which rep is which) and rescaling (because the Casimir itself is only well-defined up to rescaling).

Under this assignment we have $\mathfrak{sl}_2$ and $\mathfrak{so}_3$ going to different points. How is this possible? My best guess is that $\mathfrak{sl}_2$ and $\mathfrak{so}_3$ are different as metric Lie algebras, but that also seems weird.

In the conventions of this paper $\mathfrak{sl}_2$ corresponds to the point $(-1:1:1)$ while $\mathfrak{so}_3$ corresponds to the point $(-1:2:-1)$ and these are different points in $\mathbb{P}^2/S_3$. You can also easily check that the points are different in other convetions.

This question was originally asked in comments by Scott Carnahan, but I wanted to move it up to the main page.