Your frame is always important, and you always have some frame...
People have different accomplishments. The guy might be 20 years older and have 100 more accomplishments than you. Ok, he is essentially on 'Alpha' position in comparison to you because you are comparing yourself to his accomplishments. So by default you might feel "lesser" than him because of his good accomplishments (more money, older, more education/knowledge, more experience,...). And true, this can be sometimes intimidating.
But if you follow Sigma frame it is much easier. You do of course respect his accomplishments, but at the same time you don't really give a damn about them. Why? Because in you mind your own accomplishments are much more important. It kind of have to do it with your self esteem: You are the high value man, and no matter who you deal with, you are still the high value man. To be that man you should develop and keep strong frame...
As Beta you naturally may feel on lesser position if all you have is $2,000 while the other guy has $100,000. As Sigma, you naturally could care less about his $100,000 because what is really important to you is those $2,000...
As Sigma, you create your own hierarchy, your own frame, your own ladder. You are at the top of your ladder, you don't step onto nor follow any other ladders. As Sigma, your $2000 are at the top of YOUR ladder, and his $100,000 are below. No, they don't even belong to your ladder, they are simply on somebody else's ladder, who cares, it is not important to you. You value your own accomplishments no matter what others accomplished. However, you don't appear arrogant because you RESPECT him and his accomplishments as well. You simply keep him on your side as a friend, as a guy who you respect... You want to have those guys around, you want to learn from them so eventually there is $100K at the top of your ladder too...
Arrogant person is different. He wouldn't respect his accomplishments, he would just claim that he is the best with his $2,000, and none of the other guy is as good as him... Which is kind of ignorant. Arrogant person kind of stands against the other guy, he minimizes the other guy's accomplishments. I wouldn't do that, I would keep the other guy on my side, I would acknowledge the other guy's accomplishments....
Another way to see it is, that you don't want to appear much "higher" than the other person. For example, he is 20 years older and has many accomplishments greater than yours. That is good, give him the respect, he deserves it. He is at the top of his ladder. But, at the same time, you don't minimize your age and your accomplishments. You are 20 years younger but you have already accomplished ABC, which is great, top of your ladder...
There are probably better ways to explain it, I just can't think about any at the moment... You simply know your value as a man, and you always keep that value. Your value should be higher than the other person's value...
For example, with girls you don't want your value to be too high so she can't reach it, but it can't be too low either because she would disrespect you. You just want to be little bit over hers. If her pedestal is 100 inches from the ground, place your pedestal 120 inches from the ground. This way she can comfortably reach it, just one step, but if you place yours 150 inches she won't be able to reach it and she will fall into auto-rejection...
Ok, so maybe that guy's pedestal is 100 inches from the ground. Maybe you want to place yours to 90 inches. This way it is pretty much equal, yet you can still respect him and learn lots of things from him. Arrogant person would probably place his own pedestal to 500 inches from the ground, totally disregarding the other guy(s)...
You can use assertiveness - he asks you to do something, ok, no big deal. But you asks for something in return. Sometimes you don't even have to, sometimes all you have to say is something like: "Ok, I will do this for you, but you owe me for this one". Normal person should get right away, usually he doesn't want to "owe" you anything, and you should make it clear that it is 50/50. Next time he asks you to do the same, so you say something: "Ok, you owe me for these two things already - what you gonna do for me if I'll do this?". That is much stronger response, now you are asking him what he's gonna do for you. Now he's got to thing harder. Third time he asks you, so you reply: "Ok, no problem, but this is third time I'm doing something for you. I'll do it but only if you do this (XYZ) for me". And that is it, you showed good will, you did something extra for the other person, but now you are also requesting compliance. You have created condition - you'll do it but only if he does something back for you first...
Groups - similar to above, there is a guy who is on "Alpha" position towards you. Sigma frame will fix it fast - who gives a damn about him? He might be a great guy who you can respect, but he's got to earn the respect first. You are not trying to prove yourself to him (nor anybody else). The opposite is true: With the right frame, he's got to prove himself to you. You join the group and have fun, you have a high value as a man, and you could care less who is Alpha or Beta in that group... Yet, at the same time, your frame is open to respect and friendliness. You are friendly to everybody in that group. If he shows to be more of an asshole, ok, perhaps laugh and mirror him. Do the same, be an asshole to him - and be friend to everybody else. If he is fake Alpha he will destroy himself soon by getting upset. If he is true Alpha, he will want you to be on his side. If he is just an asshole who currently dominates that group - well, most likely he will remain an asshole no matter what you do...