We report a multiple-gain-element Nd:YAG laser where the gain media (three pieces of slab crystal) are alternately bonded to two optical quality 4H-SiC wafers. Such composite gain configuration can efficiently remove waste heat from the gain medium, preventing thermal lensing and heat-induced birefringence/distortion under high power laser operation. Through near Brewster's angles incidence designing and polarization discrimination, two orthogonally linearly polarized (P and S polarized) laser beams are generated simultaneously from different parts of the same system. Based on a T = 3% output coupler, this continuous wave laser produces maximum power of 5.34 W (0.83 W) with a slope efficiency of 21.1% (3.6%) for the S (P) polarized laser beam. At the 5-W level, the S polarized beam has a beam quality of M2~1.2. The wavelengths of these two perpendicularly polarized laser beams differ about 0.6 nm (1063.7 and 1064.3 nm). Polarized output behavior dependent on the output-coupler transmission is also studied, and it is found that increasing the transmission leads to steady growth of the P polarized laser beam; when a T = 1.3% output coupler is adopted, more than 99% of the output is the S polarized beam. The highest total output power is 6.75 W obtained with the T = 1.3% output coupler, corresponding to slope efficiency of 25.7%. This composite laser scheme, bonding multiple gain media with high-thermal-conductivity materials, opens a new avenue for high-power high-beam-quality solid-state lasers with multiple-polarization output beams.