  We treat the lattice sine-Gordon equation and two of its generalised
symmetries as a compatible system. We use a connection with the lattice
Liouville equation to aid the construction of symmetries and establishing
relations with other integrable equations. Elimination of shifts from the two
symmetries yields an integrable NLS-type system. An auto-B\"acklund
transformation and a superposition formula for this system is obtained by
elimination of shifts from the lattice sine-Gordon equation and its
down-shifted version. We use the obtained formulae to calculate a superposition
of two and three elementary solutions.
