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The automorphisms of all 4-dimensional, real Lie Algebras are presented in acomprehensive way. Their action on the space of $4\times 4$, real, symmetricand positive definite, matrices, defines equivalence classes which are used forthe invariant characterization of the 4-dimensional homogeneous spaces whichpossess an invariant basis.Comment: LaTeX2e, 23 pages, 2 Tables. To appear in Journal of Physics A:  Mathematical & Genera

Automorphisms of real four-dimensional Lie algebras and the invariant characterization of homogeneous 4-spaces