"We sleep before we eat."
Translation:Vi sover innan vi äter.
Innan does start a subordinate clause here. The whole concept of 'inversion' isn't a good way of thinking about Swedish word order, if we have it in any of the lesson notes we should weed it out (they aren't searchable so I don't know).
The thing is this:
- In main clauses, the verb goes second
- In subclauses, the subject goes before the verb, and inte goes before the verb too.
This means that you may well have the subject before the verb in both a main clause and a subclause (but for different reasons), so it's not a good idea to call it "inversion".
So for instance in this specific case, the only difference in word order between main clause and subclause is where inte would appear if you put it in – it would be like this:
Vi sover inte innan vi Inte äter.
So as you can see, in the main clause inte would be after the verb, but in the subclause it would be before the verb. But since there isn't an inte in our sentence, both main clause and subclause get the same word order.
We don't use the present participles that way in Swedish, so that won't work either (we use them mainly as adjectives). If you really really want to skip the second subject, you'd have to switch to a noun instead: Vi sover innan maten works, which literally means 'We sleep before the food', only it sounds natural.