On locally boundedly exactly doubly transitive lie groups of transformations of the space with a subgroup of parallel translations

The paper solves the problem of extending the group of parallel translations of a three-dimensional space to a locally boundedly exactly doubly transitive group of transformations for the case of a decomposable Lie algebra. The Lie algebra of the required Lie group of transformations is represented as a semidirect sum of a commutative three-dimensional ideal and a three-dimensional Lie subalgebra. Basis operators are found for all Lie algebras of doubly transitive Lie groups of transformations with a subgroup of parallel translations. The Lie groups of transformations are restored from the basis operators.