  We consider weakly closed transitive algebras of operators containing
non-zero compact operators in real Banach spaces (Lomonosov algebras). It is
shown that they are naturally divided in three classes: the algebras of real,
complex and quaternion classes. The properties and characterizations of
algebras in each class as well as some useful examples are presented. It is
shown that in separable real Hilbert spaces there is a continuum of pairwise
non-similar Lomonosov algebras of complex type and of quaternion type.
