Factorized S-matrix and the Bethe ansatz for simple lie groups

The factorized S-matrices as well as the eigenvalues of the transfer matrices for symmetrical degrees of fundamental representations of all Lie groups are presented in the form of the Bethe ansatz in terms of roots systems. As an application the explicit expressions of the S-matrices for the lowest dimension representations of some exceptional Lie groups and the solution of the principal chiral model are calculated.