PM communication - written

A couple thoughts come to mind in addition to the excellent thoughts from the community. In no particular order...

If you can talk to people first, do that before the email. You can spend a lot of time trying to write something perfectly that you could sort out very quickly by talking to someone. If you need a follow-up email for documentation, you can then be brief and not misinterpreted.

Ask for help rather than stating demands. It provides much less opportunity for people to read your intention as hostility.

If something needs to be done ASAP, explain why there is urgency. What are the consequences if they don't? I frequently have this discussion frequently when asking for help from those above me: How hard do I have to push and why? Explain the consequences of the "do nothing solution".

Ask for when they can reasonably meet your request considering their other obligations. That shows you understand they are already busy.

When asking for input, I will often use the terms "Please advise." Please shows respect and advise shows I value their input.

I would close the email with something closer to "Your help on this is greatly appreciated." than Thanks, Best Regards, Sincerely, or something that sounds very impersonal.

Also, have a peer review it first as Thomas suggested. Things sound different in your own head than they do to others.

Write short sentences. Depending on your mother tongue you may want to write long sentences but that can be confusing.
you can write factual emails but you have to put some sentences to look not to direct.

for instance, when you begin. 'I hope you are keeping well. (you and your family for some countries).

then
Do it please. do it ASAP. looks quite direct and ordering.
Would you mind trying ....if you have time?

after you have to adapt to people you work with. Even if you are a manager, in some countries it does not mean you have the right to give orders or to be too direct.
but the more you practise the better it is.
it is also important to have regular feedback loops with people you send emails to. and you can tell people: 'I may sound direct in this email ..'. Perhaps when people do not know you it is better to build a relationship over the phone before writing emails.