Every element of the fundamental group of a connected Lie group is represented by a homomorphism from the circle group

Statement
Suppose $$G$$ is a connected Lie group. Denote by $$S^1$$ the circle equipped with the obvious group structure (e.g., the structure of multiplication as complex numbers of modulus 1), and where the basepoint is chosen as the identity element for that group structure. Consider the fundamental group $$\pi_1(G,\operatorname{id}_G)$$ of $$G$$ with basepoint its identity element, i.e., the set of based homotopy classes of based continuous maps from the pair $$(S^1,\operatorname{id}_{S^1})$$ to the pair $$(G,\operatorname{id}_G)$$.

For every element $$\alpha \in \pi_1(G,\operatorname{id}_G)$$, there exists a homomorphism of groups $$\varphi: (S^1,\operatorname{id}_{S^1}) \to (G,\operatorname{id}_G)$$ such that $$\alpha$$ is the based homotopy class of $$\varphi$$.

Related facts

 * Every element of the fundamental group of a connected topological group need not be represented by a homomorphism from the circle group