Lie's third theorem

Every finite-dimensional real Lie algebra is isomorphic to the Lie algebra of a unique connected, simply connected Lie group up to isomorphism.

Lie algebra: A vector space equipped with a bilinear, antisymmetric bracket satisfying the Jacobi identity. The theorem starts with this infinitesimal algebraic structure.
Tangent space at the identity: The tangent vector space of a Lie group at its identity element. A Lie group's Lie algebra is built on this tangent space.
Sophus Lie: A Norwegian mathematician who founded the systematic theory of continuous transformation groups. His program sought to relate infinitesimal transformations to global groups.
Integration of Lie algebras: The process of constructing Lie groups whose infinitesimal structure is a given Lie algebra. This is the existence problem that the theorem resolves for finite-dimensional real Lie algebras.
Lie group: A smooth manifold with a group structure whose multiplication and inversion are smooth. The theorem identifies the global object integrating the given Lie algebra.
Left-invariant vector field: A vector field on a Lie group unchanged by every left translation. Left-invariant fields turn tangent vectors at the identity into a global algebra of fields.
Wilhelm Killing: A German mathematician whose work helped establish the structure theory of Lie algebras. His investigations advanced the algebraic side of the theory behind Lie's result.
Baker–Campbell–Hausdorff formula: A formula expressing the logarithm of a product of exponentials in a Lie algebra using nested brackets. It gives a local multiplication law from bracket data, complementing the theorem's global existence.
Lie bracket: The bilinear operation in a Lie algebra that encodes infinitesimal commutation. The bracket is the algebraic data that the resulting group's tangent structure must preserve.
Exponential map (Lie theory): A map from a Lie algebra to its Lie group that sends infinitesimal generators to one-parameter subgroups. It provides the local link between algebra elements and group elements.

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