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
By linearity, it is enough to check the Jacobi identity on the basis elements. When there are repetitions on the Jacobi identity it is satisfied automatically. Therefore you have to check nothing!