The exhaustive analytic horizon-component case split

Either some component rotates or none does. In the second alternative all components are non-rotating; either at least one is degenerate or none is. These are mutually exclusive alternatives and yield precisely (R), (D) and (N). No assumption that every component has the same degeneracy type is introduced.

For $K^{\mathrm{\prime }}=aK$ with constant $a\mathrm{\ne }0$, the Killing-horizon equation gives ${\mathrm{\nabla }}_{K^{\mathrm{\prime }}}{K}^{\mathrm{\prime }}=a^2{\mathrm{\nabla }}_{K}K=a\kappa K^{\mathrm{\prime }}$. Thus the new coefficient is $a\kappa$, which vanishes exactly when $\kappa$ does. Also scaling the stationary field by a positive constant does not change its tangency to the same generator lines. The case partition is consequently independent of those constant normalizations.