= Casimir normalization for orthogonal vector representations
{c}
{title2=$C_2(\mathbf N)=(N-1)/2,\quad c(\mathbf N)=1$}
For the defining <representation> of the <Special orthogonal Lie algebra>, Hermitian <Lie algebra generators> $t_{ij}=i(E_{ij}-E_{ji})/\sqrt2$ have <trace> normalization $\operatorname{tr}(t_{ij}t_{kl})=\delta_{ik}\delta_{jl}$ for ordered pairs. Summing their squares gives $\sum_{i<j}t_{ij}^2=(N-1)I/2$. Consequently the <Casimir eigenvalue> is $(N-1)/2$ and the <Dynkin index> is one. Doubling the trace normalization doubles both numbers; a <gauge coupling> must be rescaled inversely with the <Lie algebra generators>. With long <roots> of squared length two, the first convention is $C_2(\lambda)=(\lambda,\lambda+2\rho)/2$.
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