The discretized Bethe-Salpeter eigenvalue problem arises in the Green’s function evaluation in many body physics and quantum chemistry.  Discretization leads to a matrix eigenvalue problem for H of order 2n with a Hamiltonian-like structure. After an appropriate transformation of H to a standard symplectic form, the structure-preserving doubling algorithm, originally for algebraic Riccati equations, is extended for the discretized Bethe-Salpeter eigenvalue problem. Potential breakdowns of the algorithm, due to the ill condition or singularity of certain matrices, can be avoided with a double-Cayley transform or a three-recursion remedy. A detailed convergence analysis is conducted for the proposed algorithm, especially on the benign effects of the double-Cayley transform. Numerical results are presented to demonstrate the efficiency and the structure-preserving nature of the algorithm.