On the nested algebraic Bethe ansatz for spin chains with simple g-symmetry

We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with g-symmetry for any simple Lie algebra g. Starting the nesting process by removing a single simple root from g, we use the residual $U(1)$ charge and the block Gauss decomposition of the R-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.