Let ω=e^2π i/7, and define α=ω+ω^2+ω^4 and β=ω^3+ω^5+ω^6.

Since $1+\omega+\dots+\omega^6=0$, the six non-trivial powers add to $-1$, so $\alpha+\beta=-1$ (A true). Expanding $\alpha\beta$ gives nine terms $\omega^{a+b}$; exactly three of them have $a+b\equiv0\pmod 7$ and contribute $1$ each, while the other six are the six distinct non-trivial powers, summing to $-1$. So $\alpha\beta=3-1=2$. Hence $\alpha,\beta$ are roots of $x^{2}-(\alpha+\beta)x+\alpha\beta=x^{2}+x+2$ (B true), giving $\alpha,\beta=\frac{-1\pm i\sqrt7}{2}$. Then $|\alpha|^{2}=\frac{1+7}{4}=2$, so $|\alpha|=\sqrt2\neq2$ (C false). Because $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}>0$, $\alpha$ is the root with positive imaginary part, $\operatorname{Im}\alpha=\frac{\sqrt7}{2}$ (D true). A numerical evaluation confirms all of this. Answer: A, B, D.