# Classification of low-dimensional real Lie algebras

In mathematics, there is a classification of low-dimensional real Lie algebras.

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] Below we give Mubarakzyanov's classification[1] and numeration of these algebras. For review see also Popovych et al.[2] For each algebra ${\displaystyle {\mathfrak {g}}}$ we adduce only non-zero commutators between basis elements.

## Contents

### One-dimensional

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

### Two-dimensional

• ${\displaystyle 2{\mathfrak {g}}_{1}}$, abelian;
• ${\displaystyle {\mathfrak {g}}_{2.1}}$, solvable,
${\displaystyle [e_{1},e_{2}]=e_{1}.}$

### 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 VIII, ${\displaystyle {\mathfrak {so}}(3),}$
${\displaystyle [e_{2},e_{3}]=e_{1},\quad [e_{3},e_{1}]=e_{2},\quad [e_{1},e_{2}]=e_{3}.}$

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 ${\displaystyle \beta \rightarrow \infty }$, 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.