  The structure of subring $J^{\Gamma \cap \Gamma^{-1}}$ of the asymptotic
Hecke algebra is described for $\Gamma$ a left cell of the Coxeter group of
type $H_4$. A small set of generators is produced. The subalgebras spanned by a
subset of the basis ${t_x}_{x\in \Gamma\cap\Gamma^{-1}}$ are determined.
