Lemma: Affine current difference

The distribution $B_{\rho \mu }$ is antisymmetric, and

Proof. The first term is $(X_{\rho }{\partial }_{\mu }L-X_{\mu }{\partial }_{\rho }L)/3$; the second is antisymmetric because $X$ is Killing. Thus $B_{\rho \mu }=-B_{\mu \rho }$.

Moreover,

Since ${\partial }_{\rho }{X}_{\mu }$ is constant and ${\partial }_{\rho }{X}^{\rho }=0$, differentiating (eq:affine-superpotential) gives

The derivative of the second term cancels the last expression.