It’s always nice when new grammar is actually just old grammar being recycled for new purposes.
And I’ll Form The Verb!
Blue: The います in て います is just the polite form of いる.
Blue: That means you can use other いる conjugations to make other sorts of “-ing” sentences.
Yellow: Like using plain いる to be casual. “I am writing” as 書いている instead of 書いています.
Blue: Sometimes people don’t pronounce the い in the casual form, so it can also be 書いてる.
Yellow: And I guess you can talk about the past too. Like “I was writing” with書いていました.
Blue: And of course you could make a casual past “-ing” sentence too.
Blue: So by combining the て conjugation and いる conjugation rules we get a new set of conjugations.
Yellow: It would be cooler if they combined into some sort of giant grammar robot.