The speciality index and the Lifshitz-Khalatnikov Kasner index parametrization

The speciality index function ${\mathcal S}$ for any Petrov type I, II or D spacetime is shown to be a natural function of a single complex scalar quantity $\mu$ (natural modulo permutation symmetries). For the family of Kasner spacetimes, this quantity is a function of the Kasner indices alone which coincides with the real Lifshitz-Khalatnikov parameter $u$ for those indices.