 Casimir normalization for orthogonal vector representations

For the defining representation of the Special orthogonal Lie algebra, Hermitian Lie algebra generators $t_{ij}=i(E_{ij}-E_{ji})/\sqrt{2}$ have trace normalization $\mathrm{tr}(t_{ij}{t}_{kl})={\delta }_{ik}{\delta }_{jl}$ 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 $C_2(\lambda )=(\lambda ,\lambda +2\rho )/2$.