Accurate simulations of complex multibody systems with both rigid and flexible components are the basis of advanced controller design. Lie group variational integrators exhibit significant advantages in long-term energy conservation. Projective geometric algebra (PGA) represents geometric elements (points, lines, planes) and rigid body motions in vector form, and provides a compact representation for Lie group operations. In this paper, we integrate these two powerful tools by constructing the Lie group variational integrator fully based on the PGA, without any matrix operation. We also introduce new formulas for the differential of the exponential map, and model holonomic constraints with infinite stiffness. Numerical experiments validate the computational efficiency of the proposed new formulas, and simulations of three typical mechanisms demonstrate the long-term energy conservation of the integrator and its high-order convergence rate.