  Like all smooth manifolds, a Lie group G has a tangent vector space attached to each element. In particular, the tangent space at the neutral element 0 of G becomes the Lie algebra of G by the definition of the so-called Lie bracket, a bilinear mapping of T0G × T0G into T0G, which, for all u,v,w in T0G satisfies the condition [u,u] = 0 and the Jacobi identity: [u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0. The Lie algebra of G throws much light on the structure of G through the homeomorphic (“exponential”) mapping of a neighborhood of 0 ∈ T0G into a neighborhood of 0 ∈ G. 