  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.
