Partnership is why we ballroom dance. The feeling of being "one" with your partner is like no other feeling. You and your partner become a single entity (a four legged animal). The frame is one of the most important elements of the four legged animal (partnership). The two legged animals (partners) are joined by the frame to create the complete four legged animal.
In International Standard style of dancing the three points of connection of the frame stay in contact for the whole dance except maybe in the very beginning of the dance. In American Smooth style of dancing you may have one, two, three or no connection of the frame during the different parts of the dance.
The frame is everything, without the frame there is no partnership. The frame is just as important whether it is connected by one side, two sides or neither side. The frame is always there no matter what kind of connection or no physical connection you have to your partner.
Even with no physical contact with your partner, there is a visual, logical and a balance connection. The free arm of a fencer is just as active as the foil arm. With single and couples ice skaters, both arms are involved all of the time. Both arms of the high wire walker are very involved as are both arms of a ballerina dancer.
"For each action there is an equal and opposite reaction" (whether you are two legged or four legged). This pretty well describes what the frame is all about. Many dancers do not use the free arm when it is not connected to their partner. The free arm must not only remain active but it must continue to be a participating part of the partnership.
You hear a lot about "arm styling". I don't believe there is such a thing. The so called arm styling is actually how the free arm or arms should react as part of the four legged animal. Some instructors teach that the arms should do such and such. I believe that the intent is that this is how the arms act as part of the partnership. Most students take the arm movements as just something to do with their arms instead if realizing that this arm action should be a part of the natural free movement of the partnership.
Just as you use both legs all of the time when moving, you use both arms all of the time no matter what the position of your frame. There is as much energy and connection in a free arm of your frame as there is in a connected arm. Every part of the two legged animal is always connected to the other side of the four legged animal whether connected physically or logically. All parts of the frame are always there.
Although we are referencing Standard and Smooth dancing, it is just as important in Latin and Rhythm dancing. The free arm/arms give you continuation of movement, centrifical force, leverage, power, connection through the connected arm, counter balancing, logical and visual partnership, etc. The two sides of the four legged animal need as much connection as possible to maintain the "oneness" of the single entity.
Please don't loose the partnership feeling by having a non-existent free arm frame. The free arm frame is a very important ingredient of the total four legged animal. If you work on your so called "free arms" as always being an active part of the partnership, your dancing will improve more than you can imagine.