Left-invariant Riemannian metrics on Lie groups have provided examples of distinguished metrics, such as Einstein and Ricci soliton. One of central questions in this area is to examine whether given Lie groups admit distinguished left-invariant metrics or not. The answer is well understood for three-dimensional unimodular Lie groups, for which the Milnor frames play fundamental roles. Recently, we have developed a general procedure to obtain a generalization of Milnor frames for any Lie groups, not only in dimension three. The method is based on the study of the space of left-invariant metrics on a Lie group, which is relevant to submanifold geometry in noncompact symmetric spaces. In this talk, we explain the procedure we have developed, and describe some explicit examples. We will also mention that our procedure can also be applied to left-invariant pseudo-Riemannian metrics. This talk is based on several joint works with Takahiro Hashinaga, Akira Kubo, Kensuke Onda, Yuichiro Taketomi, and Kazuhiro Terada.