= Solution
Let $B$ denote the zero-diagonal matrix with off-diagonal entries appearing in the iteration, and $G=-B/\alpha$ its <iteration matrix>. Its row sums are three, so $B\mathbf1=3\mathbf1$. The other eigenvalues are $2\omega+\omega^2$ and $2\omega^2+\omega$, where $\omega=e^{2\pi i/3}$, namely $-3/2\pm i\sqrt3/2$, both of modulus $\sqrt3$. Thus the <spectral radius> is $\rho(G)=3/|\alpha|$.
For $|\alpha|>3$, the infinity-norm estimate $\|G\|_\infty=3/|\alpha|<1$ already gives a contraction, so the iterates converge from every initial vector to the unique solution of $(\alpha I+B)x=b$. Conversely, two iterates whose initial difference is a nonzero multiple of $\mathbf1$ retain the difference $(-3/\alpha)^k\mathbf1$. This fails to decay when $|\alpha|\le3$, so convergence cannot be guaranteed for arbitrary starts. Therefore
$$
\boxed{|\alpha|>3.}
$$
At $\alpha=3$ the offending mode oscillates; at $\alpha=-3$ it is stationary and the coefficient matrix is singular. A specially chosen starting vector can suppress a bad mode, but that is not guaranteed convergence.
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