Joint optimization for common rate allocation and beamforming design have been widely studied in rate splitting multiple access (RSMA) empowered multiuser multi-antenna transmission networks. Due to the highly coupled optimization variables and non-convexity of the joint optimization problems, emerging algorithms such as weighted minimum mean square error (WMMSE) and successive convex approximation (SCA) have been applied to RSMA which typically approximate the original problem with a sequence of disciplined convex subproblems and solve each subproblem by an optimization toolbox. While these approaches are capable of finding a viable solution, they are unable to offer a comprehensive understanding of the solution structure and are burdened by high computational complexity. In this work, for the first time, we identify the optimal beamforming structure and common rate allocation for the weighted sum-rate (WSR) maximization problem of RSMA. We then propose a computationally efficient optimization algorithm that jointly optimizes the beamforming and common rate allocation without relying on any toolbox. Specifically, we first approximate the original WSR maximization problem with a sequence of convex subproblems based on fractional programming (FP). Numerical results show that the proposed algorithm achieves the same performance but takes only 0.5\% or less simulation time compared with the state-of-the-art WMMSE, SCA, and FP algorithms. Experimental results show that the proposed two methods have similar mean square error (MSE) performance compared with traditional method using CVX optimization tool, however, computational complexities are greatly reduced. The proposed algorithms pave the way for the practical and efficient optimization algorithm design for RSMA and its applications in 6G.