Nested algebraic Bethe ansatz for orthogonal and symplectic open spin chains

We present a nested algebraic Bethe ansatz for one-dimensional so2n- and sp2n-symmetric open spin chains with diagonal boundary conditions. The monodromy matrix of these spin chains satisfies the defining relations on the extended twisted Yangians X_\rho(so2n; so2n^\rho)^tw and X_\rho(sp2n; sp2n^\rho)^tw, respectively. We use a generalisation of the De Vega and Karowski approach allowing us to relate the spectral problem of so2n- or sp2n-symmetric open spin chain to that of gln-symmetric open spin chain studied by Belliard and Ragoucy. We explicitly derive the structure of Bethe vectors, their eigenvalues and the nested Bethe equations. We also provide a proof of Belliard and Ragoucy's trace formula for Bethe vectors of gln-symmetric open spin chains.