Refinement of Ado's Theorem in Low Dimensions and Application in Affine Geometry

In this article, we construct a faithful representation with the lowest dimension for every complex Lie algebra in dimension ≤ 4. In particular, in our construction, in the case that the faithful representation has the same dimension of the Lie algebra, it can induce an étale affine representation with base zero which has a natural and simple form and gives a compatible left-symmetric algebra on the Lie algebra. Such affine representations do not contain any nontrivial one-parameter subgroups of translation.