Generalized Milnor's frames of some 4-dimensional metric Lie algebras

We consider a procedure to obtain Milnor-type theorems ([1]) by constructing a special orthonormal frames (Milnor's frames) of metric Lie algebras in dimension 4. This method is based on the results [2], devoted to the study of spaces of left-invariant Riemannian metrics on Lie groups. These frames are useful for computing the invariant tensor fields on Lie groups with left-invariant Riemannian metrics.
References: [1] J. Milnor, "‘Curvatures of left invariant metrics on Lie groups"’, Adv. Math. 21(3), 293– 329 (1976). [2] H. Kodama, A. Takahara, H. Tamaru, "‘The space of left-invariant metrics on a Lie group up to isometry and scaling"’, Manuscripta Math. 135, 229–243 (2011).