The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a group action on a smooth manifold.
The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group.
The equivalence between the category of simply connected real Lie groups and finite-dimensional real Lie algebras is usually called (in the literature of the second half of 20th century) Cartan's or the Cartan-Lie theorem as it was proved by Élie Cartan.
[2] Ado's theorem states that any finite-dimensional Lie algebra can be represented by matrices.
A more geometric proof is due to Élie Cartan and was published by Willem van Est [nl].