Inside ideas: Prodigy Math, Khan Academy, free language learning apps like Duolingo from the library, Quizlet flashcards for stuff he should memorize, handwriting practice, even more audiobooks. More active ideas: running laps, going through a series of exercises (maybe printed on cards, different each day), weeding the garden, washing the cars, Wii Fit or whatever is popular now, video workouts, vacuuming, scrubbing the tub. I would be hesitant to add too much computer time for a kid who is already disruptive, it tends to make it worse.
But I would also be working towards him being able to be alone for short periods of time, and not audibly complaining every time he is bored. Some kids really need consistent training for this type of thing; have you tried anything in particular? It sounds like he might need a lot more vigorous exercise for starters, and probably exercise breaks as often as every hour. I'm assuming this is the youngest, listed as 7 in your siggy?
When we did habit training when my kids were younger, we would do things like have jewels or coins in jar, or tickets. Focus on one thing at a time, either complaining OR occupying himself, not both at once. You give up a jewel, coin, or ticket when you complain or interrupt or whatever. You might get an extra token for doing particularly well. Tokens are then traded in for small treats or privileges: a special snack, a visit to the park, a toy, whatever. Slowly build up to longer periods of time and fewer (but bigger) rewards.
Does he get enough time out of the house, with other people? He might need more stimulation and company if you are usually home. A few random ideas:
*Bring the two younger ones to a park day while the teen works at home or in a coffeeshop nearby. This doesn't help the problem of working with the brother alone directly, but adds stimulation and active play to the schedule.
*Split them up during the day - have one in a ten o'clock activity and the other in an eleven o'clock activity, you can work with each one alone while the other is in the activity. Try to find another mom to trade playdates with, particularly for the 8-yr-old. Look for a mother's helper to come in once or twice a week.
*If they have a positive relationship, can the teen throw a ball or run laps with the 8-yr-old while you work with his brother? A bit of bribery might help here.
*Homeschooling is very flexible; work with them one-on-one in the evenings or on weekends, if that's when you have another person in the house. You will be amazed at how much you can get done with no interruptions, or fear of interruptions!
*Start giving him an explicit idea of time. Practice waiting quietly for two minutes, five minutes, ten minutes. Have visually obvious clocks and timers around the house and in the schoolroom - no, you have not been waiting forever, you have been waiting seven minutes. Countdown timers can be good because they show how long is left.
*Add visual reminders to not interrupt when you are working with brother. You can buy an array of stop signs at the educational store, lol. Have a visual timer on the table; every time he interrupts or disrupts with noise, take away a token PLUS add five minutes to the timer. He has to know that behavior is only going to delay the attention he wants. This works especially well if you give a lesson before something he definitely does not want to delay, like lunch or a trip to the park.
*Switch it around and work with his brother first. If you can, let the 8-yr-old sleep later and get stuff done before he is even up.
*Get some cheap webcams and toss him outside. Seriously, particularly if you have a fenced-in backyard, toss him outside with a glass of water and shut the door so you can't hear the groaning. You can keep an eye on him via webcam.
*Valued activities should be kept for after school only - after school meaning when they are BOTH done. For a lot of kids, this will be screen time, which is another reason I would hesitate to add even educational screen time to the school day. If you want school to be done by 2 o'clock, then free choice screen time is 2 to 2.30. If your disruptions mean everyone isn't finished until 2.15, oh well, you just lost half of your screen time for the day.
You have to be calm and consistent. You have to be relentless. But it can be done. He can be alone, and he can leave his brother's lessons alone. It just has to be easier to go ahead and do that than to disrupt.