  There are six different mathematical formulations of the symmetry group in
quantum mechanics, among them the set of pure states $\mathbf{P}$ -- i.e. the
set of one-dimensional projections on a complex Hilbert space $H$ -- and the
orthomodular lattice $\mathbf{L}$ of closed subspaces of $H$. These six groups
are isomorphic when the dimension of $H$ is $\geq 3$. The latter hypothesis is
absolutely necessary in this identification. For example, the automorphisms
group of all bijections preserving the orthogonality and the order on
$\mathbf{L}$ identifies with the bijections on $\mathbf{P}$ preserving
transition probabilities only if dim$(H)\geq 3$. %(see [G. Cassinelli et al.,
\emph{Rev. Mat. Phys.}, 1997]). This identification leads to Uhlhorn's version
of Wigner's theorem. Despite of the difficulties caused by $M_2(\mathbb{C})$,
rank two algebras are used for quantum mechanics description of the spin state
of spin-$\frac12$ particles, there is a counterexample for Uhlhorn's version of
Wigner's theorem for such state space.
  In this note we prove that in order that the description of the spin will be
relativistic, it is not enough to preserve the projection lattice equipped with
its natural partial order, but we also need to preserve the partial order set
of all tripotents, which strictly enlarges the set of projections. Concretely,
we show that a bijection between the sets of tripotents of two atomic
JBW$^*$-triples non-containing rank-one Cartan factors, preserving the natural
order in both directions and satisfying some weak continuity hypothesis can be
extended to a complex-linear or to a conjugate-linear triple automorphism.
This, in particular, extends a result of Moln{\'a}r to all bounded symmetric
domains associated with Cartan factors of rank $\geq 2$ and atomic
JBW$^*$-triples non-containing rank-one Cartan factors, and provides new models
to present quantum behavior.
