I am the leader of a tech team. One of my colleagues, who was only one grade lower than me, not only satirized and antagonized me in public but also questioned or satirized me in the work chat group for many times. The first time things happened, I expressed the view to my immediate superior that the employee was provoking the relationship between me and the team, but my superior said that as long as he worked well, it would not be a problem. I also try to be tolerant or patient, because he has a potent ability to work, and I don’t worry about giving the work to him; I don’t have enough confidence that my superior will support me to deal with him. Finding superiors to support this behavior may not only be unable to achieve desired results but also affect the leaders’ views on me.
Your manager has told you clearly that you work in a company where "he is nasty and upsets people but he gets a lot done" is one of their guidelines. Companies generally don't change this rule. I recommend finding a company that doesn't nurture nastiness. Nice people can also get a lot done when they're not being mocked and belittled.
If you think that perhaps this isn't what the whole company is like, just your manager, you could discuss it more firmly with your manager, or even the next manager up. I would only do that if I was otherwise determined to leave, as a last attempt to find a way to stay. I think word gets around pretty quickly in companies and if in general they don't keep nasty people even though they are smart and productive, then a single team where the nasty person is being kept attracts attention. Someone tells the manager to do something about their toxic team member. That's not happening and I think it's not happening for a reason. Believing that "but he gets so much done" is more important than "he is horrible to others" and not realizing how much those others could get done without the nastiness is a common attitude throughout many technical companies.
According to what you wrote in your question, I am inclined to say to go to your manager, and if your manager doesn't cooperate, then escalate to HR as this may constitute bullying and/or workplace harassment. However, one of the things in the comments caught my eye:
I am working in an eastern country. I can provide an example here, once a time I posted an algorithm improvement suggestion comment in group chat, he asked me not to post this until after I tried and proved successful.
This is not "satirizing" or "antagonizing" you. This is asking you to show your work. "Satirizing" you would be like, I'm guessing from your name that you are Asian, like pulling your eyelids out and putting on a fake Chinese accent and mimicking something you said like that. Antagonizing you would be aggressively saying something criticizing your work or you directly in front of others, like "DengDeng's code is always broken why can't he do anything right?". Neither of those are acceptable behaviour anywhere, but much less in a workplace environment.
This is not what happened. You were neither satirized nor antagonized. Your coworker saw a suggestion you made, and he asked you to show your work by proving that what you said actually works. Being generous to your coworker, I believe there is a context that is missing from the story you are telling (because discussions like this almost never happen between reasonable people without mitigating context, and I believe most people are reasonable), my guess as to the context is the following (in decreasing order of probability):
You made a suggestion for this algorithm, and your coworker believes it doesn't work. (I'd put this probability at about 90%, in that I believe it is almost definitely true (that your coworker believes your suggestion doesn't work; I have no knowledge about your suggestion, maybe it works and maybe it doesn't))
1, and also your coworker explained to you why it wouldn't work, and you wouldn't accept their reasoning, and/or that the mistake in the algorithm is so obvious that someone of your level shouldn't need it explained to them, and in either case that your coworker is telling you to go implement it so you will find the flaw yourself. (I'd put this probability at about 75%, in that I believe it is fairly likely)
2, and also you have a history of doing this and tend to do it often, where you suggest code that doesn't work, and then have to have it explained to you why it doesn't work, and your coworkers are fed up with explaining to you why your code is bad. (I'd put this probability around 40%, in that I believe it is plausible but probably not true, although it happens more than you might expect)
(Adding to my belief of this context is the fact that you claimed to be "antagonized" and "satirized", which are actually pretty inflammatory accusations, and then after a little bit of probing it is obvious that neither of those actually happened, so I am less likely to believe the rest of your story at face value and more likely to believe that mitigating factors exist)
Now, as for what you should do: I can't tell you what context you missed in this discussion, because I don't have the conversation in front of me. You should go back and read the discussion again, and see if you can figure out which of the above happened. If, after rereading the conversation, you are still not able to ascertain the context in which this comment was made, you may want to present the conversation to your manager and ask them for feedback. Perhaps a second pair of eyes may help to elucidate the context. If your manager, after having seen the conversation, is unable to ascertain the context, then you can ask your manager to do something about it, if you think what happened was disrespectful. However, when you present the content of the conversation to your manager, be sure to not edit out any portion of it; any part you edit out "because it's unnecessary" or whatever might contain valuable information to help your manager help you ascertain the context so you can improve your relations with your coworkers, and that's the ultimate goal. Also, when you send the conversation content to your manager, emphasize to your manager what the purpose is: You feel you were mistreated, but you may have missed some context and need help ascertaining it and you want to improve your relationship with your team. If you frame it as "my team was rude to me and I want you to take action against them" and then your manager turns out to believe that your team was right and you were wrong, you may find yourself having made the situation worse.
Your manager should be responsible for sorting this out, but it sounds like he isn't interested or aware of how this type of thing impacts a business.
I'm assuming you have no official authority to sort this out yourself. You could have a word in private with your colleague, explain how this is undermining your authority and ability to do you job, and ask them to stop.
If they don't, then you could make an official complaint to your HR department (or boss if you don't have HR).
Before doing this, do some research about workplace bullying and harassment, and see what the the employment laws are in your country regarding this.