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.
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.