  We propose a basis for rational gl(N) spin chains in an arbitrary rectangular
representation $(S^A)$ that factorises the Bethe vectors into products of
Slater determinants in Baxter Q-functions. This basis is constructed by
repeated action of fused transfer matrices on a suitable reference state. We
prove that it diagonalises the so-called B-operator, hence the operatorial
roots of the latter are the separated variables. The spectrum of the separated
variables is also explicitly computed and it turns out to be labelled by
Gelfand-Tsetlin patterns. Our approach utilises a special choice of the spin
chain twist which substantially simplifies derivations.
