= Killing-normalized SU(2) quadratic Casimir
{c}
{title2=$C_\kappa=\tfrac12j(j+1)I$}
With Hermitian angular-momentum generators $[J_a,J_b]=i\epsilon_{abc}J_c$, the <Killing form> is $2\delta_{ab}$ and its inverse contraction is $C_\kappa=\tfrac12\sum_aJ_a^2$. Equivalently the anti-Hermitian basis $T_a=-iJ_a$ has <Killing form> $-2\delta_{ab}$ and gives the same operator. In the complex highest-weight basis, $C_\kappa=H^2/8+(E^+E^-+E^-E^+)/4$. It acts on spin $j$ by $j(j+1)/2$, so it differs in normalization from the usual angular-momentum <quadratic Casimir operator>.
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