The actions of the triceps and biceps muscles at the shoulder joint are secondary to those at the elbow joint. Because of their attachment to the scapula, when doing elbow flexion or extension exercises, the muscles of the shoulder joint must contract to hold the shoulder and arm in place, i.e., to be stabilized.
If not, the muscles will have a tendency to perform their actions at both the elbow and shoulder joints. A multitude of muscles come into play both for shoulder stability and to allow for a well-executed movement at the elbow joint. The flexors of the elbow joint, the biceps brachii, brachialis, brachioradialis and pronator teres are involved most when you use a neutral or supinated grip. With the neutral grip, when you execute elbow flexion, the pronator teres cancels out the tendency of the biceps to supinate the hand so that the hand remains in the neutral position during execution.
This is an example of working synergy. Because the brachialis attaches to the ulna bone, it is equally strong, regardless of the grip being used. This is not true of the biceps brachii.
The two heads of the biceps cross the shoulder joint to attach on the scapula. However, their action at the shoulder joint is relatively weak, and they come into play only when the resistance is sufficiently great. At this time they act mainly as secondary movers and help stabilize the shoulder joint.
Since only the long head of the triceps crosses the shoulder joint it plays a role as a stabilizer but even more importantly, as a prime mover for shoulder joint extension. Thus it plays a key role not only at the shoulder but at the elbow. The rotator cuff muscles handle most of the stabilization work on the posterior shoulder.