  We unfold the theta integrals defining the Kudla-Millson lift of genus 1
associated to even lattices of signature (b,2), where b>2. This enables us to
compute the Fourier expansion of such defining integrals and prove the
injectivity of the Kudla-Millson lift. Although the latter result has been
already proved by Bruinier, our new procedure has the advantage of paving the
ground for a strategy to prove the injectivity of the lift also for the cases
of general signature and of genus greater than 1.
