Why to use lower case "1080p" and "1080i" but not capital "1080P" and "1080I"?
Here are my starting premises.

In the scientific community, There are some standards in the assignation of letters for units.
Normally a capital letter is assigned if the unit cames after the last name of a person, like Kelvin or Newton, and a lowercase letter in the case of just words like the case of a meter.

In the case of the liter, where it is not based on the last name of a person, the usage of capital L is permitted to avoid confusion with the number 1.

There are also some conventions on the usage of acronyms, where you should use Upper Case letters, for example, PPI.

And there are some more on the usage of abbreviates, where lower case letters should be used.



My assumption is that convention 1 and 2 are the most important in this case.   Although in this case, they are not units, the reasoning could apply.
p and i, in my opinion, are not acronyms, so the capital letter would not override the first 2 conventions. They can also derive just from an abbreviation so premise 4 would also apply.

Regarding the K in a 4K resolution, according to one note in the wikipedia it is just an informal way to say thousand, but with an implied unit, in this case the implied unit could be a pixel, 4kpx. It is based on the idea of the kilo but does not use the standard notation.
But something curious is that it is the common practice to denote a Kilobyte and not kilobyte. Probably to make it more uniform with some other prefixes in Uppercase, like Megabyte and Terabyte.
One idea about prefixes was that lowercase prefixes were used to define smaller than the base unit, and Uppercase to define greater than the unit... but the kilogram has a really strange history. Add that the unit for temperature is a big K.
It could also be just a gimmick. We were used to see the capital M (based on the standard prefixes) regarding resolution, for example, &quot;a 24Mpx camera&quot;. Using a lower case letter probably has less impact than using a capital letter.
&quot;Goodbye to the old 1080p, Wellcome the 4k&quot;
&quot;Goodbye to the old 1080p, Wellcome the 4K&quot;