Lie algebraic noncommuting structures from reparametrisation symmetry

We extend our earlier work of revealing both space-space and space-time noncommuting structures in various models in particle mechanics exhibiting reparametrisation symmetry. We show explicitly (in contrast to the earlier results in our paper \cite{sg}) that for some special choices of the reparametrisation parameter $\epsilon$, one can obtain space-space noncommuting structures which are Lie-algebraic in form even in the case of the relativistic free particle. The connection of these structures with the existing models in the literature is also briefly discussed. Further, there exists some values of $\epsilon$ for which the noncommutativity in the space-space sector can be made to vanish. As a matter of internal consistency of our approach, we also study the angular momentum algebra in details.