Many cases can cause red eyes after workout. Commonly, most of bloodshot is resulted from dry eyes and broken blood vessel of eyes.
In a gym with air-condition, your eyes focus on coach and your body. You forget to blink eyes and the airflow evaporate tears quickly. To prevent dry eyes, drink plenty of water before and during workout.
Broken blood vessel of eyes is subconjuctival hemorrhage. Certain exercise, like lifting weights, will increase blood pressure and burst tiny blood vessels in the white of eyes. You do not worry about, the redness will be absorbed by body. Within a few days, it will be ok.