= Solution
The <characteristic polynomial> of a real $2\times2$ <matrix> is $p(z)=z^2-Tz+D$. Thus $p(1)=1-T+D$ vanishes exactly when $T=D+1$, and $p(-1)=1+T+D$ vanishes exactly when $T+D+1=0$. These are the required <eigenvalue> conditions.
For $D=1$ and $|T|<2$, the <discriminant> is negative, and
$$
z_\pm=\frac T2\pm i\sqrt{1-\frac{T^2}{4}}.
$$
Their moduli equal one. Taking $\alpha\in(0,\pi)$ with $\cos\alpha=T/2$ gives \b[$z_\pm=e^{\pm i\alpha}$]. The strict <trace> inequalities exclude $\alpha=0,\pi$ and hence exclude the repeated real multipliers $+1,-1$.
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