What Is a Lie Algebra?

This paper explores Lie algebras as the infinitesimal counterparts of Lie groups, which are smooth manifolds equipped with a compatible group structure. After recalling fundamental notions of groups, manifolds, and the origins of Lie theory, we view the tangent space at the identity as a linear approximation and pose the central question: when a manifold carries a group structure, what algebraic structure does its tangent space inherit? This leads naturally to the concept of the Lie bracket. Focusing on matrix groups, we introduce linear groups and show how the commutator bracket arises from group multiplication via the exponential map. To illustrate the geometric intuition, we include figures that highlight the role of the tangent plane. We then present the formal definition of a Lie algebra and demonstrate how it encodes the local symmetries of its parent group, supplemented by illustrative examples. The paper culminates in a concise sketch of the proof of Lie's Third Theorem, which asserts that every finite-dimensional Lie algebra is associated with some Lie group, thereby completing the bridge between abstract theory and applications.