Jacobi convergence for a three-by-three equicorrelation matrix

For a three-by-three matrix with diagonal entries one and every off-diagonal entry $\alpha$, the Jacobi method iteration matrix has eigenvalue $-2\alpha$ on the all-ones vector and eigenvalue $\alpha$ with multiplicity two on its orthogonal complement. Its spectral radius is $2|\alpha |$, so Jacobi iteration converges exactly when $|\alpha |<1/2$.