In this paper we consider the regularized version of the Jacobi algorithm, a block coordinate descent method for convex optimization with differentiable objective function and block-separable constraints that has been recently proposed in the literature. Under certain regularity assumptions on the objective function, this algorithm has been shown to satisfy the so-called sufficient decrease condition, and consequently to converge in objective function value. In this paper we revisit the convergence analysis of the regularized Jacobi algorithm and show that it also converges in iterates under very mild conditions on the objective function. Moreover, we establish conditions under which the algorithm achieves a linear convergence rate.