I dug into the literature but could not find references for some of the basic H-space properties of $SO(3)$. Basic properties that I am looking for include

• What H-maps are there $SO(3)\rightarrow S^3$?
• For what integer $N$ is the $N^{th}$ power map $N:SO(3)\rightarrow SO(3)$ an H-map?
• What is the group $[SO(3)\times SO(3),SO(3)]$?
• What is the known about the commutator $c:SO(3)\wedge SO(3)\rightarrow SO(3)$?

If anyone knows where calculations of these things - or any other relevant interpreting tidbits - may be found it would be very welcome. I was surprised not to find any of this basic information.