Locally symmetric left invariant Riemannian metrics on 3-dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3-dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E-0(2) is the only 3-dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.