Lie algebra and Lie groups are closely related for example by exponentiation map.
Given the commutators of Lie algebra generators, one can compute the multiplications between Lie group elements by Baker Hausdorff formula. So we get group multiplication rules.
For discrete groups, like cyclic group, knowing the multiplication rules means knowing the structure of the group. For Lie algebra, knowing commutator relations or structure constants are enough for the structure of the Lie algebra.
The Jacobi identity is a feature of Lie algebra. Giving some constraints on the structure constants of Lie algebra.
I am wondering what does it mean for the corresponding Lie group. Is there any formula meaningful as the exponentiation of Lie algebra generators?