Throwing a Football: Both Arms Are Important
There is obviously a ton of emphasis put on the throwing arm when training a quarterback how to properly throw a football but what often gets forgotten is the non-throwing arm. Now I know it shouldn't be emphasized as much as the dominate arm but that does not mean it does not play a significant role in properly throwing a football. When a quarterback pulls their non-throwing arm across their body and then backwards during their throwing motion, it helps rotate their shoulders and generates more torque through the torso, which improves the velocity of the ball. This technique for the non-throwing arm also pulls their shoulders back, improving his/her posture and force production.
Using Drew Brees as an example, you can see how he uses his left arm to start the rotation of his shoulders. Also notice when he rips his left arm across his body and then backward, it pulls his shoulder blades together and gives him that improves posture or "big chest". This technique creates a bunch of torque through his midsection and allows him to big his right arm through with more force.
Here is a simple drill from the Sean Shelton Quarterback Academy camp to help improve this technique.