Where do these L brackets go for an LG Dual Inverter install?

The manual instructions for that LW8022IVSM model acutally specify the **inner windowsill**, not the outer one, which is likely why you can't find a logical spot for them. You're supposed to find the center point of your window and then space those L-brackets about 8 inches out to the left and right. They sit right on top of the sill with the vertical part of the bracket flush against the back edge of the wood or vinyl frame. Those brackets aren't really designed to hold the weight of the unit or prevent it from falling out the back—your external support bracket and the window sash handle that. The real purpose of these little L-shaped pieces is to act as a physical stop. They keep the bottom of the AC from sliding forward into the room and help lock in that slight rearward tilt needed for proper drainage. Since you already have a seprate support bracket installed, the safety risk is pretty much non-existent. You coud skip them if your sill shape makes it impossible, but I'd suggest finding a way to secure the base regardless. These dual inverter units are vibration-resistant, but they aren't vibration-free. Over a long summer, a unit can slowly shift its way out of position if the bottom isn't anchored to something. If you can't get the brackets to fit, just double-check that the unit is tilted back about a quarter-inch. If it sits perfectly level, the condensate won't drain correctly. If the unit is already bolted or screwed into your external support, you're fine to leave the L-brackets in the box. Most of the time, the window sash coming down into the top channel provides enough downward pressure to keep everthing stable anyway.

That makes way more sense than what I was trying to do outside. If those brackets are just there to stop sliding, could I use some heavy-duty foam or double-sided mounting tape instead? I'm really hesitant to drill directly into my window frame if I don't have to. Also, does skipping the brackets affect how much the unit hums? I got this specific model becasue everyone says it's super quiet, and I'm worried about any rattling if it's not bolted down perfectly.

Avoid the tape. The adhesive will melt and slip the second the sun beats on that sill for a few hours. If you're skipping the brackets, the hum usually comes from the unit's base vibrating against the stool. I've found that a strip of heavy-duty EPDM rubber under the unit does a better job than the brackets for noise dampening anyway. The Dual Inverter logic is great, but because it cycles through diferent RPMs, it might hit a resonant frequency that makes your window sash rattle. If you hear a buzz, just jam some shims or foam in the side tracks to keep the cabinet tight.

I've got the 10k version of that unit. Honestly, the biggest thing with these inverters is making sure the window sash itself is locked down tight. Even if you skip the L-brackets, if that window can wiggle, the frame might rattle when the compressor kicks into high gear. I used some cheap screw-in sash locks on the tracks to keep the window pressed hard against the top rail. It stops the window from being nudged up from outside and kills vibration. As long as the external bracket holds the weight and the sash is locked, you're golden.

One thing to watch if you skip the L-brackets is the side curtain frames. On these LGs, if the base isn't fixed, those plastic accordion panels can bow out or rattle when the compressor ramps up. I noticed my unit was creeping toward the room about a millimeter a week just from the fan motor torque. Since you have a support bracket, you're safe from it falling, but check those side gaps frequently. If the unit shifts and they open up, you'll lose your efficiency. A small bead of removable caulk along the side panels can help keep things airtight if you relly hate the brackets.