Let us equip ${\bf M}_2({\mathbb R})$, a $4$-dimensional space, with the standard operator norm. Consider the unit sphere $S$, which is homeomorphic to $S^3$. It contains ${\bf O}_2$, which is the disjoint union of two circles, ${\bf O}_2^+$ (the rotations) and ${\bf O}_2^-$ (the symmetries).

Show that ${\bf O}_2^+$ and ${\bf O}_2^-$ are linked.