The book "Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces" offers a comprehensive examination of differential geometry in the context of Lie groups and Lie algebras. It highlights new geometric problems that have arisen in connection with subsymmetric spaces and the concepts of mirrors. The work is based on the research of L.V. Sabinin and specifically addresses the classification of tri-symmetric spaces generated by commuting subsymmetries. This monograph is of interest not only to mathematicians but also to physicists dealing with the unitary symmetry of elementary particles. The concepts discussed in this book are fundamental for understanding the geometric structure of homogeneous Riemannian spaces and their symmetries. The detailed analysis and theoretical foundations make this work a valuable resource for professionals and students in the fields of mathematics and physics.