When Is a Sum of Partial Reflections Equal to a Scalar Operator?

We describe the set $\widetilde{W}_n$ of values of the parameter $\alpha\in\mathbb{R}$ for which there exists a Hilbert space $H$ and $n$ partial reflections $A_1,\dots,A_n$ (self-adjoint operators such that $A_k^3=A_k$ or, which is the same, self-adjoint operators whose spectra belong to the set $\{-1,0,1\}$) whose sum is equal to the scalar operator $\alpha I_H$.