On automorphisms of finite von Neumann algebras leaving invariant some reflexive lattices

For a fixed double triangle lattice in a finite factor, we show that the subgroup of automorphisms of the factor leaving invariant the reflexive lattice generated by the double triangle lattice is isomorphic to a closed subgroup of SO(3). In particular, if the nontrivial projections in the lattice is free, then the group is explicitly determined as the symmetric group of 3 elements.