  A chiral coordinate Bethe ansatz method is developed to study the periodic
XYZ chain. We construct a set of chiral vectors with fixed number of kinks. All
vectors are factorized and have simple structures. Under roots of unity
conditions, the Hilbert space has an invariant subspace and our vectors form a
basis of this subspace. We propose a Bethe ansatz solely based on the action of
the Hamiltonian on the chiral vectors, avoiding the use of transfer matrix
techniques. This allows to parameterize the expansion coefficients and derive
the homogeneous Bethe ansatz equations whose solutions give the exact energies
and eigenstates. Our analytic results agree with earlier approaches, notably by
Baxter, and are supported by numerical calculations.
