  Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be
locally isometrically embedded into a sufficiently high dimensional Euclidean
space. However, for an individual Riemannian manifold (M,g), it is in general
hard to determine the least dimensional Euclidean space into which (M,g) can be
locally isometrically embedded, even in the case where (M,g) is homogeneous. In
this paper, when the space (M,g) is locally isometric to a three-dimensional
Lie group equipped with a left-invariant Riemannian metric, we classify all
such spaces that can be locally isometrically embedded into the
four-dimensional Euclidean space. Two types of algebraic equations, the Gauss
equation and the derived Gauss equation, play an essential role in this
classification.
