Lie's third theorem - that every finite-dimensional Lie algebra (over the reals) is the tangent space of some Lie group - is quite difficult to establish without first establishing Ado's theorem that every finite-dimensional Lie algebra embeds into the Lie algebra of a general linear group. But once Ado's theorem is in place, the claim is quite easy: the Lie group is the space of curves in the general linear group that are tangent to the embedded linear algebra, up to smooth deformation.

(Without Ado's theorem, it is relatively straightforward to make the Lie algebra the tangent space of a local Lie group via the Baker-Campbell-Hausdorff formula, but establishing the global associative law necessary to extend this local Lie group to a global one is highly non-trivial. The main contribution of Ado's theorem, then, is allowing one to exploit the global associativity of the general linear group, which is a triviality in this concrete setting, but not in the setting of the abstract local group given by BCH.)

Lie's third theorem - that every finite-dimensional Lie algebra (over the reals) is the tangent space of some Lie group - is quite difficult to establish without first establishing Ado's theorem that every finite-dimensional Lie algebra embeds into the Lie algebra of a general linear group. But once Ado's theorem is in place, the claim is quite easy: the Lie group is the space of curves in the general linear group that are tangent to the embedded linear algebra, up to smooth deformation.

(Without Ado's theorem, it is relatively straightforward to make the Lie algebra the tangent space of a local Lie group via the Baker-Campbell-Hausdorff formula, but establishing the global associative law necessary to extend this local Lie group to a global one is highly non-trivial. The main contribution of Ado's theorem, then, is allowing one to exploit the global associativity of the general linear group, which is a triviality in this concrete setting, but not in the setting of the abstract local group given by BCH.)