I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their *Lie algebras*. More precisely, does anyone know of a text that contains proofs of statements below (modulo some assumptions):

If G is a compact, simply-connected Lie group, then its (finite dimensional) representations are in correspondence with those of the complexification of its Lie algebra;

If H is a (compact) Lie group with the same Lie algebra as G, then it is a quotient of G by a subgroup of the centre; the (finite dimensional) representations of H are precisely those for G that factor through the quotient.

I understand the representation theory of (finite-dimensional, complex, semisimple) Lie algebras, and have a (working) knowledge of differential geometry and algebraic topology; references that only consider matrix Lie groups are not preferred, though it would be nice if any particularly high-powered differential geometry\topology is kept to a minimum.