We propose a way to separate variables in a rational integrable gl(n) spin chain with an arbitrary finite-dimensional irreducible representation at each site and with generic twisted periodic boundary conditions. Firstly, we construct a basis that diagonalises a higher-rank version of the Sklyanin B-operator; the construction is based on recursive usage of an embedding of a gl(k) spin chain into a gl(k+ 1) spin chain which is induced from a Yangian homomorphism and controlled by dual diagonals of Gelfand–Tsetlin patterns. Then, we show that the same basis can be equivalently constructed by action of Bäcklund-transformed fused transfer matricies, whence the Bethe wave functions factorise into a product of ascending Slater determinants in Baxter Q-functions. Finally, we construct raising and lowering operators—the conjugate momenta—as normal-ordered Wronskian expressions in Baxter Q-operators evaluated at zeros of B—the separated variables. It is an immediate consequence of the proposed construction that the Bethe algebra comprises the maximal possible number of mutually commuting charges—a necessary property for Bethe equations to be complete.