Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative $H$-spaces in the sense that any map $f\colon X\to Y$ to a homotopy associative, homotopy commutative $H$-space extends to a uniquely determined $H$-map $\overline{f}\colon U(X)\to Y$. Developing a method for recognising certain universal spaces, we show the existence of the universal space $F_2(n)$ of a certain three-cell complex $L$. Using this specific example, we derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the $d_1$-differential of the EHP-spectral sequence valid in a certain range.