  In this paper we show that some 3-dimensional isometry algebras, specifically
those of type I, II, III and V (according Bianchi's classification), can be
obtained as expansions of the isometries in 2 dimensions. It is shown that in
general more than one semigroup will lead to the same result. It is impossible
to obtain the algebras of type IV, VI-IX as an expansion from the isometry
algebras in 2 dimensions. This means that the first set of algebras has
properties that can be obtained from isometries in 2 dimensions while the
second set has properties that are in some sense intrinsic in 3 dimensions. All
the results are checked with computer programs. This procedure can be
generalized to higher dimensions, which could be useful for diverse physical
applications.
