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.
