This mathematics-related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras, published in Russian in 1963.^[1] 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.^[2] in 2003.

Mubarakzyanov's Classification

Let ${\displaystyle {\mathfrak {g))_{n))$ be ${\displaystyle n}$-dimensional Lie algebra over the field of real numbers with generators ${\displaystyle e_{1},\dots ,e_{n))$, ${\displaystyle n\leq 4}$.^{[clarification needed]} For each algebra ${\displaystyle {\mathfrak {g))}$ we adduce only non-zero commutators between basis elements.

One-dimensional

• ${\displaystyle {\mathfrak {g))_{1))$, abelian.

Two-dimensional

• ${\displaystyle 2{\mathfrak {g))_{1))$, abelian ${\displaystyle \mathbb {R} ^{2))$;
• ${\displaystyle {\mathfrak {g))_{2.1))$, solvable ${\displaystyle {\mathfrak {aff))(1)=\left\((\begin{pmatrix}a&b\\0&0\end{pmatrix))\,:\,a,b\in \mathbb {R} \right\))$,

${\displaystyle [e_{1},e_{2}]=e_{1}.}$

Three-dimensional

• ${\displaystyle 3{\mathfrak {g))_{1))$, abelian, Bianchi I;
• ${\displaystyle {\mathfrak {g))_{2.1}\oplus {\mathfrak {g))_{1))$, decomposable solvable, Bianchi III;
• ${\displaystyle {\mathfrak {g))_{3.1))$, Heisenberg–Weyl algebra, nilpotent, Bianchi II,

${\displaystyle [e_{2},e_{3}]=e_{1};}$

• ${\displaystyle {\mathfrak {g))_{3.2))$, solvable, Bianchi IV,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}$

• ${\displaystyle {\mathfrak {g))_{3.3))$, solvable, Bianchi V,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}$

• ${\displaystyle {\mathfrak {g))_{3.4))$, solvable, Bianchi VI, Poincaré algebra ${\displaystyle {\mathfrak {p))(1,1)}$ when ${\displaystyle \alpha =-1}$,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}$

• ${\displaystyle {\mathfrak {g))_{3.5))$, solvable, Bianchi VII,

${\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2},\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}$

• ${\displaystyle {\mathfrak {g))_{3.6))$, simple, Bianchi VIII, ${\displaystyle {\mathfrak {sl))(2,\mathbb {R} ),}$

${\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}$

• ${\displaystyle {\mathfrak {g))_{3.7))$, simple, Bianchi IX, ${\displaystyle {\mathfrak {so))(3),}$

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2},\quad [e_{1},e_{2}]=e_{3}.}$

Algebra ${\displaystyle {\mathfrak {g))_{3.3))$ can be considered as an extreme case of ${\displaystyle {\mathfrak {g))_{3.5))$, when ${\displaystyle \beta \rightarrow \infty }$, forming contraction of Lie algebra.

Over the field ${\displaystyle {\mathbb {C} ))$ algebras ${\displaystyle {\mathfrak {g))_{3.5))$, ${\displaystyle {\mathfrak {g))_{3.7))$ are isomorphic to ${\displaystyle {\mathfrak {g))_{3.4))$ and ${\displaystyle {\mathfrak {g))_{3.6))$, respectively.

Four-dimensional

• ${\displaystyle 4{\mathfrak {g))_{1))$, abelian;
• ${\displaystyle {\mathfrak {g))_{2.1}\oplus 2{\mathfrak {g))_{1))$, decomposable solvable,

${\displaystyle [e_{1},e_{2}]=e_{1};}$

• ${\displaystyle 2{\mathfrak {g))_{2.1))$, decomposable solvable,

${\displaystyle [e_{1},e_{2}]=e_{1}\quad [e_{3},e_{4}]=e_{3};}$

• ${\displaystyle {\mathfrak {g))_{3.1}\oplus {\mathfrak {g))_{1))$, decomposable nilpotent,

${\displaystyle [e_{2},e_{3}]=e_{1};}$

• ${\displaystyle {\mathfrak {g))_{3.2}\oplus {\mathfrak {g))_{1))$, decomposable solvable,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{1}+e_{2};}$

• ${\displaystyle {\mathfrak {g))_{3.3}\oplus {\mathfrak {g))_{1))$, decomposable solvable,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2};}$

• ${\displaystyle {\mathfrak {g))_{3.4}\oplus {\mathfrak {g))_{1))$, decomposable solvable,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=\alpha e_{2},\quad -1\leq \alpha <1,\quad \alpha \neq 0;}$

• ${\displaystyle {\mathfrak {g))_{3.5}\oplus {\mathfrak {g))_{1))$, decomposable solvable,

${\displaystyle [e_{1},e_{3}]=\beta e_{1}-e_{2}\quad [e_{2},e_{3}]=e_{1}+\beta e_{2},\quad \beta \geq 0;}$

• ${\displaystyle {\mathfrak {g))_{3.6}\oplus {\mathfrak {g))_{1))$, unsolvable,

${\displaystyle [e_{1},e_{2}]=e_{1},\quad [e_{2},e_{3}]=e_{3},\quad [e_{1},e_{3}]=2e_{2};}$

• ${\displaystyle {\mathfrak {g))_{3.7}\oplus {\mathfrak {g))_{1))$, unsolvable,

${\displaystyle [e_{1},e_{2}]=e_{3},\quad [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2};}$

• ${\displaystyle {\mathfrak {g))_{4.1))$, indecomposable nilpotent,

${\displaystyle [e_{2},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}$

• ${\displaystyle {\mathfrak {g))_{4.2))$, indecomposable solvable,

${\displaystyle [e_{1},e_{4}]=\beta e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3},\quad \beta \neq 0;}$

• ${\displaystyle {\mathfrak {g))_{4.3))$, indecomposable solvable,

${\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};}$

• ${\displaystyle {\mathfrak {g))_{4.4))$, indecomposable solvable,

${\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{2},e_{4}]=e_{1}+e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}$

• ${\displaystyle {\mathfrak {g))_{4.5))$, indecomposable solvable,

${\displaystyle [e_{1},e_{4}]=\alpha e_{1},\quad [e_{2},e_{4}]=\beta e_{2},\quad [e_{3},e_{4}]=\gamma e_{3},\quad \alpha \beta \gamma \neq 0;}$

• ${\displaystyle {\mathfrak {g))_{4.6))$, indecomposable solvable,

${\displaystyle [e_{1},e_{4}]=\alpha e_{1},\quad [e_{2},e_{4}]=\beta e_{2}-e_{3},\quad [e_{3},e_{4}]=e_{2}+\beta e_{3},\quad \alpha >0;}$

• ${\displaystyle {\mathfrak {g))_{4.7))$, indecomposable solvable,

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};}$

• ${\displaystyle {\mathfrak {g))_{4.8))$, indecomposable solvable,

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=(1+\beta )e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=\beta e_{3},\quad -1\leq \beta \leq 1;}$

• ${\displaystyle {\mathfrak {g))_{4.9))$, indecomposable solvable,

${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{1},e_{4}]=2\alpha e_{1},\quad [e_{2},e_{4}]=\alpha e_{2}-e_{3},\quad [e_{3},e_{4}]=e_{2}+\alpha e_{3},\quad \alpha \geq 0;}$

• ${\displaystyle {\mathfrak {g))_{4.10))$, indecomposable solvable,

${\displaystyle [e_{1},e_{3}]=e_{1},\quad [e_{2},e_{3}]=e_{2},\quad [e_{1},e_{4}]=-e_{2},\quad [e_{2},e_{4}]=e_{1}.}$

Algebra ${\displaystyle {\mathfrak {g))_{4.3))$ can be considered as an extreme case of ${\displaystyle {\mathfrak {g))_{4.2))$, when ${\displaystyle \beta \rightarrow 0}$, forming contraction of Lie algebra.

Over the field ${\displaystyle {\mathbb {C} ))$ algebras ${\displaystyle {\mathfrak {g))_{3.5}\oplus {\mathfrak {g))_{1))$, ${\displaystyle {\mathfrak {g))_{3.7}\oplus {\mathfrak {g))_{1))$, ${\displaystyle {\mathfrak {g))_{4.6))$, ${\displaystyle {\mathfrak {g))_{4.9))$, ${\displaystyle {\mathfrak {g))_{4.10))$ are isomorphic to ${\displaystyle {\mathfrak {g))_{3.4}\oplus {\mathfrak {g))_{1))$, ${\displaystyle {\mathfrak {g))_{3.6}\oplus {\mathfrak {g))_{1))$, ${\displaystyle {\mathfrak {g))_{4.5))$, ${\displaystyle {\mathfrak {g))_{4.8))$, ${\displaystyle {2{\mathfrak {g))}_{2.1))$, respectively.