I'm trying to show that the extension $1 \to \{\pm 1\} \to 2I \to I \to 1$ does not split. For that, I think it's sufficient to show that $I$ is not isomorphic to any subgroup of $2I$ (?). But I'm not sure how to because I could not find any explicit list of the elements of $I$.
Wikipedia gives a list of the elements of $2I$ in terms of quaternions but there is no Wikipedia page for $I$. I, however, know that $I \cong A_5$. So, at least, is there any direct way to show that $2I$ has no subgroup isomorphic to $A_5$? Say, we're given the list of elements of $2I$ in terms of quaternions.