Milnor's construction

Question

In the Milnor's construction, $$ \mathcal{J}(G):=\underrightarrow{\lim}G^{*(k+1)} $$ where $G$ is a topological group.
I know that there is a natural freely (right) $G$-action on $\mathcal{J}(G)$ and $\mathcal{J}(G)$ is a $G$-$CW$ complex. But I dont't know how to show that $$ p:\mathcal{J}(G)\longrightarrow \mathcal{J}(G)/G $$ has an equivariant local trivialization. In other words, it is a principal $G$-bundle. Another observation is :$\mathcal{J}(G)$ is weakly contractible. If we can prove that $p$ is a principal $G$-bundle, then we obtain the fact that $p$ is a universal principal $G$-bundle by using the characteristic theorem of universal principal $G$-bundle and this complete the proof of Milnor's construction.