I would like to remove the labels "Social", "Promotions", "Updates", and "Forums", from the Gmail compose window.
They're not labels per se, but inbox categories. They're put into your labels list as a convenience, so that moving messages into or out of a category is easier. Moving a message into or out of a category informs Gmail how to categorize similar messages in the future.
You can turn them on or off in
Settings > Inbox > Categories.
Unfortunately, you can't remove them from the labels list. Probably because even though you may not be using/showing them, Gmail still uses them. (This is true even when using an alternative Inbox option, like "Priority Inbox".
All messages are automatically labeled for the first 5 categories even if you don't show the categories in your inbox.
So, short of some possible HTML/CSS trickery in your local browser, you cannot remove these categories from the labels list. At least they're sorted to the bottom.
What I do is wait till I've got a page full of emails (100 in my setup), then select All emails, then in the Labels button, page down to the end of the labels that you've applied yourself. Untick the Social, Promotions etc boxes (fortunately always at the end of the list) and you're done. DON'T FORGET to select NONE afterwards, of course, to clear the active boxes next to your emails. Sounds ponderous but it literally takes 10 seconds and you only have to do it ~twice a month.