Summary: The space \(\Omega(G)\) of all based loops in a compact simply connected Lie group \(G\) has an action of the maximal torus \(T \subset G\) (by pointwise conjugation) and of the circle \(S^{1}\) (by rotation of loops). Let \(\mu\colon : \Omega(G) \to (\mathfrak{t} \times i\mathbb{R})^{*}\) be a moment map of the resulting \(T \times S^{1}\) action. We show that all levels (that is, pre-images of points) of \(\mu\) are connected subspaces of \(\Omega(G)\) (or empty). The result holds if in the definition of \(\Omega(G)\) loops are of class \(C^{\infty}\) or of any Sobolev class \(H^{s}\), with \(s \geq 1\) (for loops of class \(H^{1}\) connectedness of regular levels has been proved by Harada, Holm, Jeffrey, and the author in [3]).