Classification of low-dimensional real Lie algebras

This mathematics-related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras, published in Russian in 1963. It complements the article on Lie algebra in the area of abstract algebra.
An English version and review of this classification was published by Popovych et al. in 2003.

Mubarakzyanov's Classification

Let be -dimensional Lie algebra over the field of real numbers
with generators,. For each algebra we adduce only non-zero commutators between basis elements.

One-dimensional

, abelian.

Two-dimensional

, abelian ;
, solvable,

Three-dimensional

, abelian, Bianchi I;
, decomposable solvable, Bianchi III;
, Heisenberg–Weyl algebra, nilpotent, Bianchi II,
, solvable, Bianchi IV,
, solvable, Bianchi V,
, solvable, Bianchi VI, Poincaré algebra when,
, solvable, Bianchi VII,
, simple, Bianchi VIII,
, simple, Bianchi IX,

Algebra can be considered as an extreme case of, when, forming contraction of Lie algebra.
Over the field algebras, are isomorphic to and, respectively.

Four-dimensional

, abelian;
, decomposable solvable,
, decomposable solvable,
, decomposable nilpotent,
, decomposable solvable,
, decomposable solvable,
, decomposable solvable,
, decomposable solvable,
, unsolvable,
, unsolvable,
, indecomposable nilpotent,
, indecomposable solvable,
, indecomposable solvable,
, indecomposable solvable,
, indecomposable solvable,
, indecomposable solvable,
, indecomposable solvable,
, indecomposable solvable,
, indecomposable solvable,
, indecomposable solvable,

Algebra can be considered as an extreme case of, when, forming contraction of Lie algebra.
Over the field algebras,,,, are isomorphic to,,,,, respectively.