The circle and line have the same Lie algebra but different Lie groups

Both groups are one-dimensional and abelian, so their tangent brackets at the identity are zero. Sending the tangent vector $1\in T_0\mathbb{R}$ to $i\in T_1{S}^1$ is therefore an isomorphism of their real Lie algebras. Concretely, the local homomorphism is $t\mapsto e^{it}$.

The circle is compact. The open cover $\{(-m,m):m\in \mathbb{N},m\ge 1\}$ of $\mathbb{R}$ has no finite subcover, so $\mathbb{R}$ is not compact. A Lie-group isomorphism is a homeomorphism and preserves compactness. Therefore the groups are not isomorphic.

In the language of [L1], both arise from the simply connected group $\mathbb{R}$: the line uses the zero kernel, while the circle uses the nonzero discrete central kernel $2\pi \mathbb{Z}$. Thus the same one-dimensional Lie algebra does not determine the connected group. The declared ${\mathsf{A}\mathsf{C}}_{\omega }$ is propagated from [L1]; the explicit witness and compactness argument introduce no additional choice.