A bound for the nilpotency of a group of self homotopy equivalences

Let $\mathcal{E}_\Omega(X)$ be the group of homotopy classes of self-homotopy equivalences of $X$ such that $\Omega f\simeq 1d_{\Omega X}$. We prove that $\mathcal{E}_\Omega(X)$ is a nilpotent group and that $\operatorname{nil} \mathcal{E}_{\Omega}(X)\le \operatorname{cat}(X)-1$.