Preliminaries -- Curvature Tensor of an Involutive Pair. Classical Inovolutive Pairs of Index 1 -- Iso-Involutive Sums of Lie Algebras -- Iso-Involutive Base and Structure Equations -- Iso-Involutive Sums of Types 1 and 2 -- Iso-Involutive Sums of Lower Index 1 -- Principal Central Involutive Automorphism of Type U -- Principal Unitary Involutive Automorphism of Index 1 -- Hyper-Involutive Decomposition of a Simple Compact Lie Algebra -- Some Auxiliary Results -- Principal Involutive Automorphisms of Type O -- Fundamental Theorem -- Principal Di-Unitary Involutive Automorphism -- Singular Principal Di-Unitary Involutive Automorphism -- Mono-Unitary Non-Central Principal Involutive Automorphism -- Exceptional Principal Involutive Automorphism of Types f and e -- Classification of Simple Special Unitary Subalgebras -- Hyper-Involutive Reconstructions of Basis Involutive Decompositions -- Special Hyper-Involutive Sums -- Notations, Definitions and Some Preliminaries -- Symmetric Spaces of Rank 1 -- Principal Symmetric Spaces -- Essentially Special Symmetric Spaces -- Some Theorems on Simple Compact Lie Groups -- Tri-Symmetric and Hyper-Tri-Symmetric Spaces -- Tri-Symmetric Spaces with Exceptional Compact Groups -- Tri-Symmetric Spaces with Groups of Motions SO(n), Sp(n), SU(n) -- Subsymmetric Riemannian Homogeneous Spaces -- Subsymmetric Homogeneous Spaces and Lie Algebras -- Mirror Subsymmetric Lie Triplets of Riemannian Type -- Mobile Mirrors. Iso-Involutive Decompositions -- Homogeneous Riemannian Spaces with Two-Dimensional Mirrors -- Homogeneous Riemannian Spaces with Groups SO(n), SU(3) and Two-Dimensional Mirrors -- Homogeneous Riemannian Spaces with Simple Compact Lie Groups of Motions G?SO(n), SU(3) and Two-dimensional Mirrors -- Homogeneous Riemannian Spaces with Simple Compact Lie Groups of Motions and Two-Dimensional Immobile Mirrors.

As K. Nomizu has justly noted [K. Nomizu, 56], Differential Geometry ever will be initiating newer and newer aspects of the theory of Lie groups. This monograph is devoted to just some such aspects of Lie groups and Lie algebras. New differential geometric problems came into being in connection with so called subsymmetric spaces, subsymmetries, and mirrors introduced in our works dating back to 1957 [L.V. Sabinin, 58a,59a,59b]. In addition, the exploration of mirrors and systems of mirrors is of interest in the case of symmetric spaces. Geometrically, the most rich in content there appeared to be the homogeneous Riemannian spaces with systems of mirrors generated by commuting subsymmetries, in particular, so called tri-symmetric spaces introduced in [L.V. Sabinin, 61b]. As to the concrete geometric problem which needs be solved and which is solved in this monograph, we indicate, for example, the problem of the classification of all tri-symmetric spaces with simple compact groups of motions. Passing from groups and subgroups connected with mirrors and subsymmetries to the corresponding Lie algebras and subalgebras leads to an important new concept of the involutive sum of Lie algebras [L.V. Sabinin, 65]. This concept is directly concerned with unitary symmetry of elementary par- cles (see [L.V. Sabinin, 95,85] and Appendix 1). The first examples of involutive (even iso-involutive) sums appeared in the - ploration of homogeneous Riemannian spaces with and axial symmetry. The consideration of spaces with mirrors [L.V. Sabinin, 59b] again led to iso-involutive sums.