How to write a letter to HR explaining the abusive language used by the reporting manager and to change my department

So, short story long I need an opinion on the step I'm gonna take which is gonna change my career and maybe my job probably if things don't turn out in my favour.

So, I recently got placed & started my first job at a product-based company on July 1st. The company specializes in making starters, electric panels, and relays from start to finish.

After the induction, I was assigned to the Production department on the 5th day of the first week in July. I was given buddies to help me through the learning phase. The buddies they are nothing but unhelpful pricks.

My manager (a little background on him he's done Diploma through mechanical and through long way of ass kissing he has become a manager in a company where he once worked as a casual and he's around 42) So my manger told me that I'll have to do night shifts from time to time after some 6 month's or so ( at that time I literally thought of quitting) but then I thought let's not, since it is close to my home and the package is decent enough. But things are turning out to be worse now, basically i would endure all of it but the reporting manager cusses a lot and i mean a lot during office hours and not because the other person has made a mistake, just aivaye... Market thoda down chali gaye le gaali, paise doob gaye thode sae le gaali, and he literally character assassinated an AM in front of the whole department (comprising of 4 AMs, Senior Engineers of their respective departments and other 3 department heads) it's a joint office rather than having us sit in different rooms since we all have to work with each other all the time.

And he does that on a daily basis like, he yells without cause (I get you are the manager you have to create that pressure) and yelling is fine but cussing someone to their face & character assassinating someone. I just can't work there anymore listening to all of that, it's starting to take a toll on me, I get home and I am frustrated af and thinking of going back there to that retched office just makes me sad like literally cried last week thinking if that were to happen to me in near future.

So, after gathering some guts and with the help of my 3 college mates I told the HR regarding this in the presece with my mates in the second month of my job. She said she'll look into it but when I didn't see any change I went back to her after 2.5 months then she said that she'll tell about it to the VP of the plant ( we had a meeting with him too, to me he looks very supportive and the guy who would listen)

She said after the 3 months, which is a review period for the employees, she'll see. So, I went today as I completed 3 months and told her about the situation. She has asked me to write an email detailing the ordeal without name calling. I just want to get my department changed, not get into any office politics with my reporting manager.

I have the recordings of him being abusive and all but I'm very confused about taking the step and how to even write this email without name calling as it might create problems for me. Please give me suggestions, any suggestions. I'm just looking for some guidance here.