  We present a nested algebraic Bethe ansatz for one-dimensional open so(2n)-
and sp(2n)-symmetric spin chains with diagonal boundary conditions and
described by the extended twisted Yangian. We use a generalization of the Bethe
ansatz introduced by De Vega and Karowski which allows us to relate the
spectral problem of a so(2n)- or sp(2n)-symmetric open spin chain to that of a
gl(n)-symmetric open spin chain. We explicitly derive the structure of Bethe
vectors and the nested Bethe equations.
