  The multi-soliton solution of the sine-Gordon equation in the presence of
elliptic-function background is derived by the inverse scattering method. The
key tool in our formulation is the Lax pair written by $4\times4$ matrix
differential operators given by Takhtadzhyan and Faddeev in 1974, which enables
us to use the conventional form of the integral representation of the Jost
solutions and Krichever's theory of commuting differential operators. As a
by-product we also provide generalized orthogonality and completeness relations
for eigenfunctions associated with indefinite inner product. The multi-soliton
solution is expressed by a determinant of theta functions and the shift of the
background lattice due to solitons is also determined using addition formula.
One kink and one breather solutions are presented by animated gifs.
