Learn Dutch: We zijn hout voor de kachel aan het kloven geweest. - We have been splitting wood for the stove.

In English, the auxiliary verb for 'to be' is 'to have' (we have been). In Dutch, the verb zijn uses itself as the auxiliary in the perfect tense. So whenever you use geweest (been), you must use a form of zijn, literally translating to 'we are been'.

Why is geweest placed after aan het kloven at the very end of the sentence?

In this perfect continuous structure, geweest is the past participle. In Dutch main clauses, the past participle is always pushed to the absolute end of the sentence. The continuous action phrase (aan het kloven) acts like a block of information that goes right before that final verb.

Does using this perfect continuous form mean we are still splitting wood?

No, the action is entirely finished. Using zijn ... aan het [verb] geweest shows that you spent a continuous, ongoing block of time doing an activity in the past, but you have stopped doing it now. If you were still doing it, you would use the present continuous: We zijn hout aan het kloven.

Could I use oven or fornuis instead of kachel for 'stove' here?