I read in book written by Karin Erdmann and Mark J. Wildon's "Introduction to Lie algebras"
"Let F be in any field. Up to isomorphism, there is a unique two-dimensional nonabelian
Lie algebra over F. This Lie algebra has a basis {x, y} such that its Lie
bracket is defined by [x, y] = x"
How to prove that Lie bracket [x,y] = x satisfies axioms of Lie algebra such that
[a,a] = 0 for $a \in L$
and satisfies jacobi identity
and can some one give me an example of two dimensional nonabelian Lie algebra
Best Answer
In $2$ dimensional case, we have $[x,y]=0$ or $[x,y]=z=ax+by$ if $a$ is zero then by changing the variables you get what you looking for but if $a$ was not zero then divide both sides by $a$ so that it becomes $$[x,y/a]=x+by/a$$ now change the $x+by/a$ variable to $z$. $$[ z-by/a,y/a]=[z,y/a]=z$$ then change $y/a$ to $u$ so that you get $[z,u]=z$