Following the approach of , the spectrum of the Hamiltonian for the XXX model
can be determined by the Bethe ansatz. In this context, for an appropriately defined family of operators  dependent on a spectral parameter  acting on the total Hilbert space  with each , a Bethe vector is a vector of the form
where .
If the  satisfy the Bethe equation
then the Bethe vector is an eigenvector of  with eigenvalue .
The family  as well as three other families come from a transfer matrix  (in turn defined using a Lax matrix), which acts on  along with an auxiliary space , and can be written as a  block matrix with entries in ,
which satisfies fundamental commutation relations (FCRs) similar in form to the Yang–Baxter equation used to derive the Bethe equations. The FCRs also show there is a large commuting subalgebra given by the generating function , as , so when  is written as a polynomial in , the coefficients all commute, spanning a commutative subalgebra which  is an element of. The Bethe vectors are in fact simultaneous eigenvectors for the whole subalgebra.