Determining the Lie Algebra of a Vector Space

[math6]

hi friends !
it is well known that a Lie algebra over K is a K-vector space g equipped
of a K-bilinear, called Lie bracket. I ask how can we determines the Lie algebra of any vector space then? For example we try the Lie algebra of horizontal space.

 

[HallsofIvy]

You seem to have the idea that there is such a thing as "the" Lie Algebra of a vector space. That's not true. There exist an infinite number of Lie Algebras for a given vector space.