The string group $String(n)$ is by definition a covering group 3-connected cover of $O(n)$ with $\pi_0=\pi_1=\pi_2=\pi_3=0$. Spin(n)$. This definition determines the homotopy type of the string group.
[In a previous version of this question I screwed up the definition and caused some confusion, see the comments below.]
A common argument is saying that "the string group cannot be a Lie group because it has vanishing $\pi_3$". This is obviously not a complete argument because $(\mathbb{R},+)$ is a nice Lie group with vanishing $\pi_3$.
What is the correct statement about Lie group structures on the string group, and how does one prove it?
