Deform a continuous mapping of $S^3$ into $G$ into a mapping into an $SU(2)$ subgroup of $G$.

Question

I am reading The Uses of Instantons (1977) by Sidney Coleman (a physics text!) and he quotes a pretty strong and useful claim:

For a general simple Lie group $G$, any continuous mapping of $S^3$ into $G$ can be continuously deformed into a mapping into an $SU(2)$ subgroup of $G$.

Question: How do you prove this claim?

In his text, Coleman attributes this result to Raoul Bott, but looking at that paper (as a physicist) the only useful thing I can find is that $$\pi_3(G)=\mathbb{Z}$$ assuming a simple, compact Lie group. Is this enough to show the claim? How do you find the homotopy?