Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I always thought that $S^0$ was the odd one out due to it not being connected however after reading page 4 of "H-spaces from a homotopy point of view" I see that it is claimed $0$-connectedness can be weakened to $\pi_0(X)$ being a group.

Does this mean the real Hopf fibration can be constructed using the Hopf construction?

If so could I have some modern references?

This question originally stemmed from the discussion here on the nforum.