1) Separate them.
2) Set up an Argue Jar with a bunch of cardboard strips or the large size tongue depressors (from the craft store) on which you have written a whole bunch of chores with a sharpie marker. Or, you can use little photos or pictures from the internet to help those who cannot read or read well. You can set up 2 jars if you think the kids are very different in their ability to do different things. Every time you hear this arguing, just tell them that play time has become arguing time, so it's time to go to the Argue Jar(s). Each child picks out (randomly or within 15 seconds - no stalling) a job to do. Ideally these will be jobs in different rooms. You know what needs to be done, but suggestions include picking up dirty clothes, putting them in the hamper, putting them in the washer (7 year old), sorting clean socks/underwear and folding washcloths (4 year old), emptying the dishwasher (except glassware), setting the table, taking out the recycling, emptying bathroom wastebaskets, cleaning the bathroom sink (and toilet if you're ambitious), dusting (provide a microfiber duster and other cleaning tools - perhaps hang a low hook in the laundry room or broom closet so they can reach things without your help), picking up toys and matching puzzle pieces, etc.
3) Take away the object(s) they are arguing about, and put it in the closet or the garage or the attic. Assign a chore price tag to that object if you want to, or a time frame for its "vacation from arguing".
4) If your daughter criticizes/corrects your son, give her the chore and not him. If he's standing up for himself, he gets 1 free pass. If he's sniping at her and escalating, then she gets 2 chores and he gets 1.
5) Don't argue with them. That creates the same nasty environment you're trying to avoid. Show them that not being mean is way more fun.
6) Understand that this will not work the first 3 times you try it. There will be more arguing. Let them know that there will be 2 chores if they keep arguing. Take a calm moment to explain the rules and the new "system" to them when there is no drama going on.
Eventually you will be able to nip this in the bud by just saying "Do you kids want to go to the Argue Jars?". You can give one warning, and then that's it - the 2nd time, there's no negotiating their way out of it.
Stay strong Mama!